Nested numerical solving problems

Scientific knowledge often takes the form of specific relationships expressed by systems of equations. For example:

• A system of ordinary differential equations connects some state variables \(x\) with some other variables \(v\) with equations with the form \(\frac{dx}{dt}= f(x, v)\).
• An algebraic equation systems says that some variables \(v\) are related so that \(f(v) = 0\)
• A differential algebraic equation system says that some state variables’ rates of change have the form \(\frac{dx}{dt}= f(x, v)\) and that they satisfy some algebraic constraints \(f(x,v)=0\).

If there is an analytic solution to the equation system, we can just include the solution in our statistical model like any other form of structural knowledge: easy! However, often we want to solve equations that are hard or impossible to solve analytically, but can be solved approximately using numerical methods.

This is tricky in the context of Hamiltonian Monte Carlo for two reasons:

1. Computation: HMC requires many evaluations of the log probability density function and its gradients.

Important

At every evaluation, the sampler needs to solve the embedded equation system and find the gradients of the solution with respect to all model parameters.

2. Extra source of error: how good of an approximation is good enough?

Reading:

• Timonen et al. (2022)
• Stan user guide sections: algebraic equation systems, ODE systems, DAE systems.

Example

We have some tubes containing a substrate \(S\) and some biomass \(C\) that we think approximately follow the Monod equation for microbial growth:

\[\begin{align*} \frac{dC}{dt} &= \frac{\mu_{max}\cdot S(t)}{K_{S} + S(t)}\cdot C(t) \\ \frac{dS}{dt} &= -\gamma \cdot \frac{\mu_{max}\cdot S(t)}{K_{s} + S(t)} \cdot C(t) \end{align*}\]

We measured \(C\) and \(S\) at different timepoints in some experiments and we want to try and find out \(\mu_{max}\), \(K_{S}\) and \(\gamma\) for the different strains in the tubes.

You can read more about the Monod equation in Allen and Waclaw (2019).

What we know

\(\mu_{max}, K_S, \gamma, S, C\) are non-negative.

\(S(0)\) and \(C(0)\) vary a little by tube.

\(\mu_{max}, K_S, \gamma\) vary by strain.

Measurement noise is roughly proportional to measured quantity.

Statistical model

We use two regression models to describe the measurements:

\[\begin{align*} y_C &\sim LN(\ln{\hat{C}}, \sigma_{C}) \\ y_S &\sim LN(\ln{\hat{S}}, \sigma_{S}) \end{align*}\]

To capture the variation in parameters by tube and strain we add a hierarchical regression model:

\[\begin{align*} \ln{\mu_{max}} &\sim N(a_{\mu_{max}}, \tau_{\mu_max}) \\ \ln{\gamma} &\sim N(a_{gamma}, \tau_{\gamma}) \\ \ln{\mu_{K_S}} &\sim N(a_{K_S}, \tau_{K_S}) \end{align*}\]

To get a true abundance given some parameters we put an ode in the model:

\[ \hat{C}(t), \hat{S}(t) = \text{solve-monod-equation}(t, C_0, S_0, \mu_max, \gamma, K_S) \]

imports

import itertools

import arviz as az
import cmdstanpy
import pandas as pd
import numpy as np

from matplotlib import pyplot as plt

Specify true parameters

In order to avoid doing too much annoying handling of strings we assume that all the parts of the problem have meaningful 1-indexed integer labels: for example, species 1 is biomass.

This code specifies the dimensions of our problem.

N_strain = 4
N_tube = 16
N_timepoint = 20
duration = 15
strains = [i+1 for i in range(N_strain)]
tubes = [i+1 for i in range(N_tube)]
species = [1, 2]
measurement_timepoint_ixs = [4, 7, 12, 15, 17]
timepoints = pd.Series(
    np.linspace(0.01, duration, N_timepoint),
    name="time",
    index=range(1, N_timepoint+1)
)
SEED = 12345
rng = np.random.default_rng(seed=SEED)

This code defines some true values for the parameters - we will use these to generate fake data.

true_param_values = {
    "a_mu_max": -1.7,
    "a_ks": -1.3,
    "a_gamma": -0.6,
    "t_mu_max": 0.2,
    "t_ks": 0.3,
    "t_gamma": 0.13,
    "species_zero": [
        [
            np.exp(np.random.normal(-2.1, 0.05)), 
            np.exp(np.random.normal(0.2, 0.05))
        ] for _ in range(N_tube)
    ],
    "sigma_y": [0.08, 0.1],
    "ln_mu_max_z": np.random.normal(0, 1, size=N_strain).tolist(),
    "ln_ks_z": np.random.normal(0, 1, size=N_strain).tolist(),
    "ln_gamma_z": np.random.normal(0, 1, size=N_strain).tolist(),
}
for var in ["mu_max", "ks", "gamma"]:
    true_param_values[var] = np.exp(
        true_param_values[f"a_{var}"]
        + true_param_values[f"t_{var}"] * np.array(true_param_values[f"ln_{var}_z"])
    ).tolist()

A bit of data transformation

This code does some handy transformations on the data using pandas, giving us a table of information about the measurements.

tube_to_strain = pd.Series(
    [
        (i % N_strain) + 1 for i in range(N_tube)  # % operator finds remainder
    ], index=tubes, name="strain"
)
measurements = (
    pd.DataFrame(
        itertools.product(tubes, measurement_timepoint_ixs, species),
        columns=["tube", "timepoint", "species"],
        index=range(1, len(tubes) * len(measurement_timepoint_ixs) * len(species) + 1)
    )
    .join(tube_to_strain, on="tube")
    .join(timepoints, on="timepoint")
)

Generating a Stan input dictionary

This code puts the data in the correct format for cmdstanpy.

stan_input_structure = {
    "N_measurement": len(measurements),
    "N_timepoint": N_timepoint,
    "N_tube": N_tube,
    "N_strain": N_strain,
    "tube": measurements["tube"].values.tolist(),
    "measurement_timepoint": measurements["timepoint"].values.tolist(),
    "measured_species": measurements["species"].values.tolist(),
    "strain": tube_to_strain.values.tolist(),
    "timepoint_time": timepoints.values.tolist(),
}

This code defines some prior distributions for the model’s parameters

priors = {
    # parameters that can be negative:
    "prior_a_mu_max": [-1.8, 0.2],
    "prior_a_ks": [-1.3, 0.1],
    "prior_a_gamma": [-0.5, 0.1],
    # parameters that are non-negative:
    "prior_t_mu_max": [-1.4, 0.1],
    "prior_t_ks": [-1.2, 0.1],
    "prior_t_gamma": [-2, 0.1],
    "prior_species_zero": [[[-2.1, 0.1], [0.2, 0.1]]] * N_tube,
    "prior_sigma_y": [[-2.3, 0.15], [-2.3, 0.15]],
}

The next bit of code lets us configure Stan’s interface to the Sundials ODE solver.

ode_solver_configuration = {
    "abs_tol": 1e-7,
    "rel_tol": 1e-7,
    "max_num_steps": int(1e7)
}

Now we can put all the inputs together

stan_input_common = stan_input_structure | priors | ode_solver_configuration

Load the model

This code loads the Stan program at monod.stan as a CmdStanModel object and compiles it using cmdstan’s compiler.

model = cmdstanpy.CmdStanModel(stan_file="../src/stan/monod.stan")
print(model.code())

functions {
  real get_mu_at_t(real mu_max, real ks, real S_at_t) {
    return (mu_max * S_at_t) / (ks + S_at_t);
  }
  vector ddt(real t, vector species, real mu_max, real ks, real gamma) {
    real mu_at_t = get_mu_at_t(mu_max, ks, species[2]);
    vector[2] out;
    out[1] = mu_at_t * species[1];
    out[2] = -gamma * mu_at_t * species[1];
    return out;
  }
}
data {
  int<lower=1> N_measurement;
  int<lower=1> N_timepoint;
  int<lower=1> N_tube;
  int<lower=1> N_strain;
  array[N_measurement] int<lower=1, upper=N_tube> tube;
  array[N_measurement] int<lower=1, upper=N_timepoint> measurement_timepoint;
  array[N_measurement] int<lower=1, upper=2> measured_species;
  vector<lower=0>[N_measurement] y;
  array[N_tube] int<lower=1, upper=N_strain> strain;
  array[N_timepoint] real<lower=0> timepoint_time;
  array[N_tube, 2] vector[2] prior_species_zero;
  array[2] vector[2] prior_sigma_y;
  vector[2] prior_a_mu_max;
  vector[2] prior_a_ks;
  vector[2] prior_a_gamma;
  vector[2] prior_t_mu_max;
  vector[2] prior_t_gamma;
  vector[2] prior_t_ks;
  real<lower=0> abs_tol;
  real<lower=0> rel_tol;
  int<lower=1> max_num_steps;
  int<lower=0, upper=1> likelihood;
}
parameters {
  vector[N_strain] ln_mu_max_z;
  vector[N_strain] ln_ks_z;
  vector[N_strain] ln_gamma_z;
  real a_mu_max;
  real a_ks;
  real a_gamma;
  real<lower=0> t_mu_max;
  real<lower=0> t_ks;
  real<lower=0> t_gamma;
  array[N_tube] vector<lower=0>[2] species_zero;
  vector<lower=0>[2] sigma_y;
}
transformed parameters {
  vector[N_strain] mu_max = exp(a_mu_max + ln_mu_max_z * t_mu_max);
  vector[N_strain] ks = exp(a_ks + ln_ks_z * t_ks);
  vector[N_strain] gamma = exp(a_gamma + ln_gamma_z * t_gamma);
  array[N_tube, N_timepoint] vector[2] abundance;
  for (tube_t in 1 : N_tube) {
    abundance[tube_t] = ode_bdf_tol(ddt, species_zero[tube_t], 0,
                                    timepoint_time,
                                    abs_tol, rel_tol, max_num_steps,
                                    mu_max[strain[tube_t]],
                                    ks[strain[tube_t]], gamma[strain[tube_t]]);
  }
}
model {
  // priors
  ln_mu_max_z ~ std_normal();
  ln_ks_z ~ std_normal();
  ln_gamma_z ~ std_normal();
  a_mu_max ~ normal(prior_a_mu_max[1], prior_a_mu_max[2]);
  a_ks ~ normal(prior_a_ks[1], prior_a_ks[2]);
  a_gamma ~ normal(prior_a_gamma[1], prior_a_gamma[2]);
  t_mu_max ~ lognormal(prior_t_mu_max[1], prior_t_mu_max[2]);
  t_ks ~ lognormal(prior_t_ks[1], prior_t_ks[2]);
  t_gamma ~ lognormal(prior_t_gamma[1], prior_t_gamma[2]);
  for (s in 1 : 2) {
    sigma_y[s] ~ lognormal(prior_sigma_y[s, 1], prior_sigma_y[s, 2]);
    for (t in 1 : N_tube){
      species_zero[t, s] ~ lognormal(prior_species_zero[t, s, 1],
                                     prior_species_zero[t, s, 2]);
    }
  }
  // likelihood
  if (likelihood) {
    for (m in 1 : N_measurement) {
      real yhat = abundance[tube[m], measurement_timepoint[m], measured_species[m]];
      y[m] ~ lognormal(log(yhat), sigma_y[measured_species[m]]);
    }
  }
}
generated quantities {
  vector[N_measurement] yrep;
  vector[N_measurement] llik;
  for (m in 1 : N_measurement){
    real yhat = abundance[tube[m], measurement_timepoint[m], measured_species[m]];
    yrep[m] = lognormal_rng(log(yhat), sigma_y[measured_species[m]]);
    llik[m] = lognormal_lpdf(y[m] | log(yhat), sigma_y[measured_species[m]]);
  }
}

Sample in fixed param mode to generate fake data

stan_input_true = stan_input_common | {
    "y": np.ones(len(measurements)).tolist(),  # dummy values as we don't need measurements yet
    "likelihood": 0                            # we don't need to evaluate the likelihood
}
coords = {
    "strain": strains,
    "tube": tubes,
    "species": species,
    "timepoint": timepoints.index.values,
    "measurement": measurements.index.values
}
dims = {
    "abundance": ["tube", "timepoint", "species"],
    "mu_max": ["strain"],
    "ks": ["strain"],
    "gamma": ["strain"],
    "species_zero": ["tube", "species"],
    "y": ["measurement"],
    "yrep": ["measurement"],
    "llik": ["measurement"]
}

mcmc_true = model.sample(
    data=stan_input_true,
    iter_sampling=1,
    fixed_param=True,
    chains=1,
    refresh=1,
    inits=true_param_values,
    seed=SEED,
)
idata_true = az.from_cmdstanpy(
    mcmc_true,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)

17:03:49 - cmdstanpy - INFO - CmdStan start processing
17:03:49 - cmdstanpy - INFO - CmdStan done processing.

Look at results

def plot_sim(true_abundance, fake_measurements, species_to_ax):
    f, axes = plt.subplots(1, 2, figsize=[9, 3])

    axes[species_to_ax[1]].set_title("Species 1")
    axes[species_to_ax[2]].set_title("Species 2")
    for ax in axes:
        ax.set_xlabel("Time")
        ax.set_ylabel("Abundance")
        for (tube_i, species_i), df_i in true_abundance.groupby(["tube", "species"]):
            ax = axes[species_to_ax[species_i]]
            fm = df_i.merge(
                fake_measurements.drop("time", axis=1),
                on=["tube", "species", "timepoint"]
            )
            ax.plot(
                df_i.set_index("time")["abundance"], color="black", linewidth=0.5
            )
            ax.scatter(
                fm["time"],
                fm["simulated_measurement"],
                color="r",
                marker="x",
                label="simulated measurement"
            )
    return f, axes

species_to_ax = {1: 0, 2: 1}
true_abundance = (
    idata_true.posterior["abundance"]
    .to_dataframe()
    .droplevel(["chain", "draw"])
    .join(timepoints, on="timepoint")
    .reset_index()
)
fake_measurements = measurements.join(
    idata_true.posterior_predictive["yrep"]
    .to_series()
    .droplevel(["chain", "draw"])
    .rename("simulated_measurement")
).copy()
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

f.savefig("img/monod_simulated_data.png")

Sample in prior mode

stan_input_prior = stan_input_common | {
    "y": fake_measurements["simulated_measurement"],
    "likelihood": 0
}
mcmc_prior = model.sample(
    data=stan_input_prior,
    iter_warmup=100,
    iter_sampling=100,
    chains=1,
    refresh=1,
    save_warmup=True,
    inits=true_param_values,
    seed=SEED,
)
idata_prior = az.from_cmdstanpy(
    mcmc_prior,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)
idata_prior

17:03:50 - cmdstanpy - INFO - CmdStan start processing
17:04:36 - cmdstanpy - INFO - CmdStan done processing.
17:04:36 - cmdstanpy - WARNING - Non-fatal error during sampling:
Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -1: 
    Exception: lognormal_rng: Location parameter is nan, but must be finite! (in 'monod.stan', line 94, column 4 to column 69)
Consider re-running with show_console=True if the above output is unclear!

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 564kB
  Dimensions:            (chain: 1, draw: 100, ln_mu_max_z_dim_0: 4,
                          ln_ks_z_dim_0: 4, ln_gamma_z_dim_0: 4, tube: 16,
                          species: 2, sigma_y_dim_0: 2, strain: 4, timepoint: 20)
  Coordinates:
    * chain              (chain) int64 8B 0
    * draw               (draw) int64 800B 0 1 2 3 4 5 6 ... 93 94 95 96 97 98 99
    * ln_mu_max_z_dim_0  (ln_mu_max_z_dim_0) int64 32B 0 1 2 3
    * ln_ks_z_dim_0      (ln_ks_z_dim_0) int64 32B 0 1 2 3
    * ln_gamma_z_dim_0   (ln_gamma_z_dim_0) int64 32B 0 1 2 3
    * tube               (tube) int64 128B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    * species            (species) int64 16B 1 2
    * sigma_y_dim_0      (sigma_y_dim_0) int64 16B 0 1
    * strain             (strain) int64 32B 1 2 3 4
    * timepoint          (timepoint) int64 160B 1 2 3 4 5 6 ... 15 16 17 18 19 20
  Data variables: (12/15)
      ln_mu_max_z        (chain, draw, ln_mu_max_z_dim_0) float64 3kB -0.4693 ....
      ln_ks_z            (chain, draw, ln_ks_z_dim_0) float64 3kB -0.06056 ... ...
      ln_gamma_z         (chain, draw, ln_gamma_z_dim_0) float64 3kB -0.3497 .....
      a_mu_max           (chain, draw) float64 800B -1.465 -2.038 ... -2.008
      a_ks               (chain, draw) float64 800B -1.201 -1.241 ... -1.25 -1.403
      a_gamma            (chain, draw) float64 800B -0.5878 -0.5633 ... -0.5415
      ...                 ...
      species_zero       (chain, draw, tube, species) float64 26kB 0.1346 ... 1...
      sigma_y            (chain, draw, sigma_y_dim_0) float64 2kB 0.1206 ... 0....
      mu_max             (chain, draw, strain) float64 3kB 0.2081 ... 0.1613
      ks                 (chain, draw, strain) float64 3kB 0.2957 ... 0.2889
      gamma              (chain, draw, strain) float64 3kB 0.532 0.5814 ... 0.58
      abundance          (chain, draw, tube, timepoint, species) float64 512kB ...
  Attributes:
      created_at:                 2024-04-29T15:04:36.501119
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • ln_mu_max_z_dim_0: 4
    • ln_ks_z_dim_0: 4
    • ln_gamma_z_dim_0: 4
    • tube: 16
    • species: 2
    • sigma_y_dim_0: 2
    • strain: 4
    • timepoint: 20
  • Coordinates: (10)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • ln_mu_max_z_dim_0
      (ln_mu_max_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_ks_z_dim_0
      (ln_ks_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_gamma_z_dim_0
      (ln_gamma_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • tube
      (tube)
      int64
      1 2 3 4 5 6 7 ... 11 12 13 14 15 16

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16])

    • species
      (species)
      int64
      1 2

      array([1, 2])

    • sigma_y_dim_0
      (sigma_y_dim_0)
      int64
      0 1

      array([0, 1])

    • strain
      (strain)
      int64
      1 2 3 4

      array([1, 2, 3, 4])

    • timepoint
      (timepoint)
      int64
      1 2 3 4 5 6 7 ... 15 16 17 18 19 20

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
             19, 20])

  • Data variables: (15)
    • ln_mu_max_z
      (chain, draw, ln_mu_max_z_dim_0)
      float64
      -0.4693 -1.504 ... -0.4682 0.5706

      array([[[-4.69256e-01, -1.50374e+00,  1.58227e+00, -2.26553e-01],
              [ 5.62989e-01,  1.48552e+00, -1.47171e+00,  1.19858e-01],
              [ 6.25408e-01,  2.18420e+00, -1.25902e+00, -3.00229e-02],
              [-1.82269e+00,  1.52901e+00, -4.23201e-01, -9.73229e-01],
              [-1.83365e-01, -1.11730e-01, -3.93457e-01, -2.63084e+00],
              [-8.26637e-01, -1.92859e-01, -1.29456e-01,  1.87292e+00],
              [ 8.30900e-01, -4.15891e-01,  3.05617e-01, -1.37699e+00],
              [-9.13624e-01, -8.82005e-02, -4.75380e-01,  1.57556e+00],
              [ 1.42216e-01, -2.16419e-01,  1.00060e+00, -1.55348e+00],
              [-2.90609e-01,  2.65098e-03, -9.65388e-01,  1.34286e+00],
              [-3.09803e-01, -1.01126e+00, -1.46333e+00, -1.59465e+00],
              [-7.28545e-01,  9.76536e-01, -5.89256e-01,  3.27855e-01],
              [ 1.67986e+00, -8.84159e-01, -6.36956e-01,  1.31198e-01],
              [ 6.05156e-01, -1.50635e+00,  3.46239e-01,  2.55973e-01],
              [-6.82700e-02,  1.67912e+00, -2.38633e-01, -1.95336e-01],
              [ 7.44719e-01, -1.04604e-01, -8.57013e-02,  6.30297e-01],
              [ 1.30059e+00, -5.26377e-01,  3.81965e-02,  2.56979e-01],
              [-5.88228e-01, -1.62895e+00,  2.05832e-01,  4.08722e-01],
              [-1.65456e+00, -1.38054e-01, -9.13256e-02,  8.98803e-01],
              [-1.17765e+00, -2.26735e+00,  1.79583e+00,  1.15659e+00],
      ...
              [ 1.40594e+00, -5.70031e-01,  1.77540e-01,  2.09888e+00],
              [-1.37362e+00,  5.76087e-01, -1.66348e-01, -2.12390e+00],
              [ 1.45261e+00, -3.75018e-01,  4.67243e-02,  2.02244e+00],
              [-1.31311e+00,  2.23282e-01,  7.01435e-02, -1.72235e+00],
              [ 4.86791e-01,  1.34436e+00, -1.15512e+00,  6.41797e-01],
              [ 7.40559e-02, -1.34106e+00,  1.66858e+00, -8.65402e-01],
              [ 2.41566e+00,  8.69685e-01, -5.87550e-01, -2.86441e-02],
              [-3.02151e+00, -4.48409e-01,  1.21846e+00, -3.95261e-01],
              [-2.89034e+00, -2.55876e-01,  1.28815e+00, -3.44323e-01],
              [-2.02451e+00, -6.67654e-01,  5.92457e-01, -4.11240e-01],
              [-1.70602e+00, -1.03732e+00,  1.42129e+00, -5.09650e-01],
              [-6.25157e-01,  1.95030e-01,  1.27813e+00,  1.90921e-01],
              [ 3.89840e-01, -4.57810e-01, -1.41708e+00, -2.94080e-01],
              [ 4.45926e-01,  3.47280e-01,  7.88391e-01,  1.12344e-01],
              [ 1.24101e+00,  2.99326e-01, -1.40466e+00, -1.19571e-01],
              [-1.41513e+00,  6.89634e-01,  4.33156e-01,  2.67484e-01],
              [ 1.33696e+00,  9.74231e-02,  1.68918e+00, -8.03215e-01],
              [ 1.23010e+00,  9.87486e-01, -9.16837e-03, -2.92470e-01],
              [-1.74356e+00, -2.90075e-01, -6.22500e-01,  7.04242e-01],
              [-1.89294e+00, -3.02392e-01, -4.68205e-01,  5.70590e-01]]])

    • ln_ks_z
      (chain, draw, ln_ks_z_dim_0)
      float64
      -0.06056 1.317 ... -0.06499 0.531

      array([[[-6.05581e-02,  1.31739e+00,  2.00306e-01, -3.66607e-01],
              [-6.28767e-02, -1.42372e+00, -2.62483e-01,  4.09297e-01],
              [-1.46289e-01, -1.72715e+00, -3.47835e-01,  6.67675e-01],
              [-1.84173e-01,  6.71338e-01,  8.11542e-01, -5.36285e-01],
              [ 1.16176e+00,  3.03402e-01,  1.52424e+00,  8.89590e-01],
              [-1.24874e+00,  2.54203e-01, -1.99091e-01, -1.73028e+00],
              [ 1.38964e+00, -2.35702e-03,  4.67006e-01,  1.39107e+00],
              [-1.55270e+00,  3.40277e-02, -3.84039e-02, -1.50175e+00],
              [ 2.12514e+00, -1.56755e-01, -3.41820e-02,  1.06241e+00],
              [-2.26157e+00,  1.21560e-01,  2.98214e-02, -9.56685e-01],
              [ 7.56616e-01, -3.72958e-01,  9.72734e-01, -1.12008e+00],
              [ 1.45851e+00, -1.62707e+00,  9.06621e-01,  9.44555e-01],
              [ 1.63405e+00,  8.18758e-01, -5.29408e-01,  1.09578e+00],
              [ 2.21051e+00,  1.06079e+00, -2.74482e-01, -1.44644e+00],
              [-2.57040e+00, -8.20089e-01,  4.77546e-01,  1.29507e+00],
              [-1.14196e+00, -2.84483e-01, -6.61431e-01,  9.77133e-02],
              [-1.50837e+00,  1.26379e+00, -4.65223e-01, -9.08702e-02],
              [ 1.30357e-01,  9.76939e-01,  1.15874e+00, -3.29643e-01],
              [ 3.93417e-01, -4.45528e-01, -2.02233e-02,  2.51486e-01],
              [ 3.77570e-02, -1.30022e-01,  9.16295e-01,  5.79956e-02],
      ...
              [-8.27449e-01,  1.17842e+00,  6.14568e-01,  5.38182e-01],
              [ 8.37021e-01, -1.12699e+00, -6.36907e-01, -5.09271e-01],
              [-9.53247e-01,  1.30758e+00,  4.30283e-01,  3.42871e-01],
              [ 1.33223e+00, -1.25909e+00, -4.39430e-01, -5.96529e-01],
              [-1.86161e-01,  9.53200e-01,  6.64381e-01,  4.86414e-01],
              [ 1.72137e-01, -7.58078e-01,  6.15101e-01, -6.87806e-01],
              [ 1.11291e+00,  4.16282e-01, -1.39740e-01, -2.53861e-01],
              [ 5.63125e-01,  1.46920e-01, -7.44105e-02,  2.21525e-02],
              [ 7.63806e-01,  4.63775e-02, -1.16626e-01, -2.87918e-01],
              [ 6.98215e-01, -5.45098e-01, -7.65742e-02,  1.89840e-01],
              [ 6.31842e-01, -1.79632e-01,  4.21378e-01, -3.08153e-01],
              [-1.83448e+00,  1.28985e+00, -3.47339e-01,  3.06023e-01],
              [ 2.25368e+00, -1.30228e+00,  5.89792e-01, -1.79583e-01],
              [-1.85373e+00,  9.28477e-01, -4.84786e-01,  7.31918e-01],
              [-1.38716e+00,  8.63660e-01,  9.47110e-02, -2.07630e-01],
              [ 1.01490e+00,  4.26556e-01, -1.00704e+00, -3.09589e-01],
              [-9.79814e-01, -1.57577e+00, -2.77420e-03, -5.33970e-01],
              [-1.18575e+00,  5.11691e-01, -6.02303e-01, -4.14954e-01],
              [ 1.46654e+00, -5.11258e-01, -2.61754e-01,  5.39492e-01],
              [ 1.72900e+00, -6.90020e-01, -6.49879e-02,  5.30991e-01]]])

    • ln_gamma_z
      (chain, draw, ln_gamma_z_dim_0)
      float64
      -0.3497 0.3664 ... 0.6056 -0.02166

      array([[[-3.49653e-01,  3.66392e-01, -1.97855e+00, -3.19036e-01],
              [ 4.20747e-01, -3.96760e-01,  1.87278e+00,  2.01343e-01],
              [ 2.46331e-01, -1.14414e-01,  1.97471e+00,  4.47199e-01],
              [ 1.57907e+00,  1.38006e+00, -1.20974e+00,  5.31892e-01],
              [-1.72291e-01,  2.57377e+00,  2.71963e-01, -5.71178e-01],
              [-1.13576e-01, -2.32883e+00, -7.98741e-02,  1.70037e+00],
              [ 3.09308e-01,  1.92574e+00,  8.32887e-01, -1.78532e+00],
              [ 8.94304e-02, -1.82395e+00, -1.11177e+00,  1.27193e+00],
              [-1.44169e-01,  1.02085e+00,  7.80011e-01, -9.97558e-01],
              [ 2.84036e-01, -9.65895e-01, -5.01993e-01,  8.27922e-01],
              [-4.75128e-01,  2.04540e+00,  2.78633e-01, -4.05433e-01],
              [-7.48884e-01, -1.11241e+00,  5.91961e-01,  1.99120e-01],
              [ 1.05717e+00, -9.05841e-01,  1.95500e+00, -8.52427e-01],
              [-2.45980e-01, -2.49122e-01,  1.11573e+00,  1.19765e-01],
              [ 2.51381e-01, -5.71827e-02, -1.09739e+00,  1.46033e-01],
              [-6.16824e-01, -8.57997e-01, -1.46637e-01, -5.22989e-01],
              [-5.92678e-01, -3.81304e-01, -4.32766e-01, -9.41726e-01],
              [ 9.12485e-01, -1.81382e-01,  4.44417e-01,  2.65647e-01],
              [ 6.41604e-01, -9.00978e-01,  3.38456e-03,  9.13106e-01],
              [-2.35133e-01, -1.33301e-01,  2.02912e-01,  2.18704e+00],
      ...
              [ 1.25629e+00, -6.57704e-01,  1.74745e+00, -4.41141e-01],
              [-1.26255e+00,  6.95547e-01, -1.72838e+00,  4.65267e-01],
              [ 9.13762e-01, -3.78973e-01,  1.77391e+00, -5.53721e-01],
              [-6.61030e-01,  5.09114e-01, -1.38021e+00,  1.45424e-01],
              [-8.58774e-01, -9.59202e-01,  5.61305e-01,  3.87074e-01],
              [-6.00229e-01,  7.84198e-01, -1.57679e+00, -8.39324e-01],
              [ 1.20723e+00,  7.75005e-01,  1.36198e+00,  1.07531e+00],
              [-1.66580e+00, -1.56325e-01, -1.25223e+00, -1.67361e-01],
              [-1.55961e+00, -8.40948e-02, -8.72067e-01, -2.95112e-01],
              [-4.74500e-01,  5.98802e-01, -5.09339e-01, -4.67376e-01],
              [-3.16043e-01, -1.37121e-01, -1.59202e+00,  2.42883e-01],
              [ 1.67394e-01, -8.96247e-01,  1.72736e+00,  8.74954e-01],
              [-7.94002e-02,  7.23969e-01, -2.03341e+00, -1.50213e+00],
              [ 1.72127e-01, -1.02348e+00,  2.05581e-01,  2.00040e-01],
              [ 1.32143e-01,  1.31891e+00,  1.88878e-01, -7.81821e-01],
              [-2.71276e-02,  5.28923e-01,  1.33039e-01,  1.02665e+00],
              [ 1.07199e+00, -3.49244e-01,  5.96780e-01, -2.19792e+00],
              [ 1.03403e+00, -1.21614e+00,  6.12552e-01, -2.37123e-01],
              [-3.80157e-01,  2.11565e+00,  7.38926e-01, -1.94275e-01],
              [-3.29097e-01,  1.84998e+00,  6.05576e-01, -2.16577e-02]]])

    • a_mu_max
      (chain, draw)
      float64
      -1.465 -2.038 ... -1.773 -2.008

      array([[-1.46536, -2.03791, -2.11855, -1.42176, -1.68283, -1.81351,
              -1.71906, -1.51965, -1.51993, -2.18449, -1.42011, -1.64089,
              -1.85065, -1.76157, -2.12171, -1.61899, -2.02914, -2.03239,
              -1.6972 , -1.56477, -1.55901, -1.69817, -1.62948, -1.7515 ,
              -1.92222, -1.643  , -1.95668, -1.90083, -1.86353, -1.84574,
              -1.66865, -1.60237, -2.06885, -1.78054, -1.99537, -1.71051,
              -1.70737, -2.04172, -1.65641, -1.77874, -1.75694, -1.99729,
              -1.85161, -1.68891, -1.69452, -1.59823, -1.52275, -1.63718,
              -1.78285, -1.83757, -1.5028 , -1.81769, -1.8259 , -1.92768,
              -1.78782, -1.81494, -1.90124, -1.57732, -2.09272, -1.76625,
              -1.56263, -1.97277, -1.68888, -1.70669, -1.7277 , -1.84565,
              -1.86091, -1.87965, -1.85814, -1.92551, -1.90998, -1.95222,
              -1.63115, -1.9153 , -1.88833, -1.69   , -1.70437, -1.76204,
              -1.83623, -1.86019, -1.56964, -2.04309, -1.5172 , -1.75583,
              -1.82458, -1.3701 , -2.09301, -2.05573, -2.17436, -1.80538,
              -1.83915, -1.87426, -1.74467, -1.61322, -1.70955, -1.85802,
              -1.68824, -1.71455, -1.77324, -2.00778]])

    • a_ks
      (chain, draw)
      float64
      -1.201 -1.241 ... -1.25 -1.403

      array([[-1.20051, -1.24142, -1.33177, -1.25284, -1.35924, -1.27624,
              -1.2676 , -1.28503, -1.31638, -1.2391 , -1.20547, -1.31259,
              -1.47234, -1.44891, -1.28186, -1.37594, -1.36452, -1.36425,
              -1.23739, -1.23739, -1.23366, -1.34155, -1.21209, -1.35298,
              -1.25789, -1.36124, -1.37842, -1.22502, -1.41079, -1.34085,
              -1.31507, -1.31758, -1.29495, -1.34831, -1.18753, -1.39337,
              -1.39123, -1.23325, -1.28491, -1.20712, -1.30598, -1.19183,
              -1.35874, -1.31249, -1.29642, -1.31469, -1.29446, -1.23535,
              -1.23827, -1.4368 , -1.31935, -1.4155 , -1.13462, -1.55136,
              -1.49181, -1.53741, -1.46577, -1.59597, -1.26928, -1.33351,
              -1.24561, -1.32536, -1.2806 , -1.30767, -1.31706, -1.26165,
              -1.26144, -1.24513, -1.23166, -1.26958, -1.29219, -1.30055,
              -1.28254, -1.29952, -1.35719, -1.31037, -1.27779, -1.18658,
              -1.41243, -1.20818, -1.31017, -1.36403, -1.4137 , -1.23465,
              -1.35325, -1.25331, -1.31863, -1.32547, -1.28635, -1.19899,
              -1.18669, -1.22602, -1.42667, -1.42093, -1.25677, -1.28345,
              -1.33378, -1.38427, -1.24954, -1.40278]])

    • a_gamma
      (chain, draw)
      float64
      -0.5878 -0.5633 ... -0.4365 -0.5415

      array([[-0.587772, -0.563285, -0.497828, -0.519355, -0.433587, -0.712169,
              -0.521935, -0.576725, -0.61767 , -0.455887, -0.462144, -0.681256,
              -0.491551, -0.50252 , -0.507413, -0.454198, -0.55703 , -0.558629,
              -0.471906, -0.38296 , -0.384276, -0.561959, -0.53707 , -0.465929,
              -0.54314 , -0.453305, -0.470582, -0.618853, -0.469858, -0.442326,
              -0.454747, -0.464361, -0.473531, -0.588951, -0.486314, -0.520447,
              -0.519092, -0.504274, -0.463393, -0.397662, -0.566397, -0.52243 ,
              -0.539027, -0.391371, -0.428903, -0.516848, -0.438159, -0.596015,
              -0.404678, -0.549471, -0.579231, -0.61211 , -0.428593, -0.525562,
              -0.418984, -0.474675, -0.38043 , -0.371504, -0.740215, -0.256692,
              -0.702254, -0.282744, -0.559665, -0.396435, -0.477675, -0.519219,
              -0.520206, -0.482585, -0.509668, -0.295527, -0.715029, -0.271177,
              -0.310721, -0.699445, -0.458744, -0.413232, -0.564239, -0.505079,
              -0.499309, -0.420312, -0.542297, -0.529121, -0.502294, -0.469921,
              -0.498653, -0.631198, -0.520826, -0.516192, -0.444328, -0.682057,
              -0.645616, -0.59335 , -0.388757, -0.421833, -0.551685, -0.461375,
              -0.636989, -0.532647, -0.436482, -0.541461]])

    • t_mu_max
      (chain, draw)
      float64
      0.2225 0.2462 ... 0.2101 0.3215

      array([[0.222451, 0.246175, 0.251114, 0.260511, 0.216407, 0.250038,
              0.255743, 0.236076, 0.226545, 0.247146, 0.245423, 0.270815,
              0.322402, 0.309433, 0.255394, 0.231533, 0.23034 , 0.230569,
              0.26291 , 0.257701, 0.256891, 0.223175, 0.225069, 0.266568,
              0.289923, 0.210057, 0.227448, 0.218848, 0.241594, 0.263396,
              0.233691, 0.235645, 0.237313, 0.262693, 0.234166, 0.267123,
              0.266449, 0.22461 , 0.311191, 0.270992, 0.216562, 0.292837,
              0.238406, 0.238508, 0.228308, 0.259638, 0.249163, 0.306936,
              0.282627, 0.234956, 0.282771, 0.240749, 0.237191, 0.225695,
              0.24932 , 0.259139, 0.24762 , 0.225649, 0.263554, 0.230312,
              0.269368, 0.23457 , 0.225297, 0.222914, 0.282407, 0.219403,
              0.219384, 0.239368, 0.251744, 0.219319, 0.278854, 0.222102,
              0.227845, 0.271248, 0.238335, 0.224974, 0.268237, 0.233719,
              0.254482, 0.310367, 0.207739, 0.205846, 0.287298, 0.257778,
              0.22774 , 0.268538, 0.238917, 0.239301, 0.243966, 0.224084,
              0.251461, 0.244698, 0.237096, 0.22725 , 0.214122, 0.272749,
              0.2183  , 0.204559, 0.210052, 0.32146 ]])

    • t_ks
      (chain, draw)
      float64
      0.2974 0.2677 ... 0.2781 0.3031

      array([[0.297397, 0.267692, 0.349681, 0.314847, 0.282779, 0.325312,
              0.317774, 0.329147, 0.316354, 0.34206 , 0.326767, 0.345031,
              0.282677, 0.26056 , 0.251925, 0.344839, 0.330978, 0.329915,
              0.272833, 0.269862, 0.269599, 0.291561, 0.292889, 0.302139,
              0.340098, 0.269196, 0.281186, 0.340235, 0.285004, 0.288361,
              0.319682, 0.319137, 0.317687, 0.297359, 0.317521, 0.263628,
              0.26369 , 0.283415, 0.376477, 0.297153, 0.288737, 0.271107,
              0.299463, 0.250287, 0.253338, 0.340523, 0.359684, 0.294335,
              0.330087, 0.322081, 0.310913, 0.350448, 0.293577, 0.301536,
              0.264508, 0.256259, 0.307714, 0.273829, 0.274754, 0.322627,
              0.303898, 0.300972, 0.300623, 0.297029, 0.277941, 0.329426,
              0.329332, 0.308383, 0.332613, 0.277584, 0.311597, 0.286712,
              0.285872, 0.325523, 0.275288, 0.270155, 0.327077, 0.315166,
              0.294945, 0.298658, 0.312973, 0.316306, 0.265169, 0.297777,
              0.318624, 0.273762, 0.2953  , 0.295066, 0.305994, 0.259673,
              0.259719, 0.309763, 0.291041, 0.291648, 0.312851, 0.286836,
              0.300187, 0.323106, 0.278123, 0.303116]])

    • t_gamma
      (chain, draw)
      float64
      0.1238 0.1353 ... 0.1148 0.1482

      array([[0.123827, 0.135292, 0.148206, 0.135298, 0.141646, 0.134367,
              0.134684, 0.140515, 0.163988, 0.164541, 0.168285, 0.115311,
              0.166307, 0.148954, 0.119431, 0.151453, 0.143081, 0.143639,
              0.129865, 0.137311, 0.137529, 0.137682, 0.146528, 0.119839,
              0.170999, 0.106613, 0.107816, 0.152089, 0.120489, 0.13189 ,
              0.151549, 0.152217, 0.135345, 0.127765, 0.147662, 0.121703,
              0.122297, 0.131504, 0.152482, 0.138612, 0.135897, 0.147743,
              0.122208, 0.125876, 0.150727, 0.11334 , 0.118656, 0.128431,
              0.137069, 0.142276, 0.124638, 0.122605, 0.158762, 0.109176,
              0.12492 , 0.126861, 0.162432, 0.133801, 0.150057, 0.119327,
              0.164156, 0.114993, 0.132584, 0.151099, 0.140538, 0.127446,
              0.127768, 0.129962, 0.141289, 0.119938, 0.145703, 0.127477,
              0.126959, 0.149616, 0.145922, 0.148726, 0.125856, 0.145454,
              0.12491 , 0.12907 , 0.140875, 0.146795, 0.140875, 0.125893,
              0.129641, 0.145372, 0.15517 , 0.155088, 0.13471 , 0.146256,
              0.149546, 0.14018 , 0.130472, 0.135005, 0.106465, 0.120821,
              0.12984 , 0.148439, 0.114773, 0.148239]])

    • species_zero
      (chain, draw, tube, species)
      float64
      0.1346 1.156 ... 0.1492 1.127

      array([[[[0.134561 , 1.15631  ],
               [0.113812 , 1.08764  ],
               [0.131563 , 1.19976  ],
               ...,
               [0.13895  , 1.14291  ],
               [0.118885 , 1.10367  ],
               [0.112284 , 1.18512  ]],
      
              [[0.149847 , 1.60001  ],
               [0.110564 , 1.2351   ],
               [0.109989 , 1.08777  ],
               ...,
               [0.123156 , 1.32953  ],
               [0.109651 , 1.11569  ],
               [0.105043 , 1.15753  ]],
      
              [[0.0972077, 0.971097 ],
               [0.137557 , 1.32654  ],
               [0.146607 , 1.32365  ],
               ...,
      ...
               ...,
               [0.113066 , 1.17425  ],
               [0.120222 , 1.36753  ],
               [0.123306 , 1.11559  ]],
      
              [[0.125266 , 1.10792  ],
               [0.145676 , 1.08141  ],
               [0.110631 , 1.2748   ],
               ...,
               [0.113635 , 1.53141  ],
               [0.130835 , 1.16636  ],
               [0.110333 , 1.35955  ]],
      
              [[0.12431  , 1.42309  ],
               [0.103455 , 1.39401  ],
               [0.13347  , 1.16174  ],
               ...,
               [0.126401 , 1.02126  ],
               [0.112083 , 1.30236  ],
               [0.149207 , 1.12726  ]]]])

    • sigma_y
      (chain, draw, sigma_y_dim_0)
      float64
      0.1206 0.1285 ... 0.09435 0.1061

      array([[[0.120565 , 0.128542 ],
              [0.0829555, 0.0745675],
              [0.10565  , 0.119102 ],
              [0.125456 , 0.111942 ],
              [0.0786671, 0.104742 ],
              [0.102748 , 0.11388  ],
              [0.0954502, 0.126566 ],
              [0.0849988, 0.0793702],
              [0.1111   , 0.12307  ],
              [0.106356 , 0.0951873],
              [0.107186 , 0.0965879],
              [0.0945894, 0.103634 ],
              [0.0789329, 0.110955 ],
              [0.0909933, 0.100319 ],
              [0.082649 , 0.113507 ],
              [0.0826568, 0.11477  ],
              [0.117029 , 0.0905379],
              [0.0946548, 0.0896989],
              [0.095481 , 0.08732  ],
              [0.0974796, 0.0898214],
      ...
              [0.112202 , 0.100018 ],
              [0.124105 , 0.0946412],
              [0.097463 , 0.0838452],
              [0.107783 , 0.0815785],
              [0.0930188, 0.123335 ],
              [0.131043 , 0.0924547],
              [0.122988 , 0.0911781],
              [0.0834241, 0.110352 ],
              [0.101938 , 0.121274 ],
              [0.0974047, 0.0808629],
              [0.0844008, 0.123137 ],
              [0.0844808, 0.106619 ],
              [0.10968  , 0.0854736],
              [0.0978241, 0.112641 ],
              [0.0977945, 0.0919136],
              [0.0858274, 0.107165 ],
              [0.100903 , 0.120367 ],
              [0.109964 , 0.0903395],
              [0.0913511, 0.111936 ],
              [0.0943478, 0.106138 ]]])

    • mu_max
      (chain, draw, strain)
      float64
      0.2081 0.1653 ... 0.1155 0.1613

      array([[[0.208097 , 0.16532  , 0.328445 , 0.219641 ],
              [0.149671 , 0.187832 , 0.0906993, 0.134203 ],
              [0.140647 , 0.208032 , 0.0876237, 0.119303 ],
              [0.150079 , 0.359357 , 0.216101 , 0.187252 ],
              [0.178616 , 0.181407 , 0.170677 , 0.105171 ],
              [0.132629 , 0.155404 , 0.157887 , 0.260486 ],
              [0.22167  , 0.161149 , 0.193805 , 0.126032 ],
              [0.176342 , 0.214281 , 0.195563 , 0.317368 ],
              [0.225889 , 0.208262 , 0.274377 , 0.153837 ],
              [0.104736 , 0.112609 , 0.0886482, 0.156828 ],
              [0.223992 , 0.188568 , 0.168766 , 0.163413 ],
              [0.159105 , 0.252479 , 0.165222 , 0.211803 ],
              [0.270073 , 0.118161 , 0.127964 , 0.163924 ],
              [0.207149 , 0.107777 , 0.1912   , 0.185933 ],
              [0.117755 , 0.183989 , 0.112741 , 0.113995 ],
              [0.235379 , 0.193359 , 0.194207 , 0.229225 ],
              [0.177362 , 0.116439 , 0.13261  , 0.139464 ],
              [0.114404 , 0.0899972, 0.13739  , 0.14397  ],
              [0.118576 , 0.176666 , 0.17885  , 0.232028 ],
              [0.154393 , 0.116592 , 0.33221  , 0.281754 ],
      ...
              [0.278713 , 0.184878 , 0.215939 , 0.321866 ],
              [0.0977009, 0.145948 , 0.125264 , 0.0837193],
              [0.332915 , 0.196923 , 0.222289 , 0.392132 ],
              [0.123154 , 0.183    , 0.175917 , 0.110824 ],
              [0.180195 , 0.219059 , 0.123979 , 0.186669 ],
              [0.259185 , 0.177245 , 0.397714 , 0.201394 ],
              [0.219617 , 0.151794 , 0.107165 , 0.122474 ],
              [0.0621143, 0.114976 , 0.171333 , 0.116448 ],
              [0.0561624, 0.106801 , 0.155658 , 0.104521 ],
              [0.104451 , 0.141566 , 0.187755 , 0.149939 ],
              [0.103504 , 0.122457 , 0.22724  , 0.139833 ],
              [0.131699 , 0.16097  , 0.20982  , 0.160808 ],
              [0.191621 , 0.156733 , 0.124849 , 0.162937 ],
              [0.220493 , 0.215605 , 0.238339 , 0.204396 ],
              [0.236024 , 0.192924 , 0.133946 , 0.176373 ],
              [0.106034 , 0.188261 , 0.175542 , 0.167786 ],
              [0.247491 , 0.188819 , 0.267272 , 0.155117 ],
              [0.231558 , 0.220347 , 0.179707 , 0.169589 ],
              [0.117716 , 0.159746 , 0.148972 , 0.19685  ],
              [0.0730737, 0.121847 , 0.115522 , 0.161321 ]]])

    • ks
      (chain, draw, strain)
      float64
      0.2957 0.4454 ... 0.2411 0.2889

      array([[[0.295669, 0.445429, 0.31952 , 0.269946],
              [0.284152, 0.197398, 0.269367, 0.322436],
              [0.250845, 0.144321, 0.233774, 0.33344 ],
              [0.269598, 0.352936, 0.368865, 0.241307],
              [0.35675 , 0.279866, 0.395258, 0.330323],
              [0.185914, 0.303144, 0.261581, 0.158956],
              [0.437793, 0.281296, 0.326542, 0.437992],
              [0.165946, 0.279758, 0.273168, 0.168752],
              [0.525145, 0.255134, 0.265221, 0.375206],
              [0.133628, 0.301942, 0.292614, 0.208807],
              [0.38357 , 0.265182, 0.411638, 0.207739],
              [0.445144, 0.153512, 0.367961, 0.372809],
              [0.364062, 0.289125, 0.197505, 0.312676],
              [0.417723, 0.30959 , 0.218619, 0.16109 ],
              [0.145235, 0.225721, 0.313001, 0.384583],
              [0.170378, 0.228998, 0.201084, 0.261258],
              [0.155089, 0.388202, 0.219041, 0.247934],
              [0.266804, 0.35277 , 0.374576, 0.229237],
              [0.323016, 0.256932, 0.288544, 0.310747],
              [0.293111, 0.280136, 0.371533, 0.294717],
      ...
              [0.208224, 0.390097, 0.326989, 0.319264],
              [0.333111, 0.178975, 0.208985, 0.217595],
              [0.188912, 0.344048, 0.27264 , 0.266393],
              [0.432601, 0.199972, 0.255253, 0.243587],
              [0.243518, 0.3501  , 0.31932 , 0.301717],
              [0.299338, 0.232041, 0.337931, 0.236549],
              [0.371582, 0.302492, 0.256688, 0.248181],
              [0.313702, 0.277449, 0.259908, 0.26742 ],
              [0.349019, 0.280226, 0.266592, 0.252979],
              [0.361431, 0.261704, 0.295562, 0.316733],
              [0.359664, 0.291318, 0.340532, 0.281755],
              [0.166248, 0.437591, 0.263524, 0.322638],
              [0.462664, 0.164362, 0.285072, 0.22788 ],
              [0.140638, 0.316592, 0.20965 , 0.298954],
              [0.184382, 0.372852, 0.29313 , 0.266674],
              [0.370707, 0.313142, 0.207566, 0.253535],
              [0.19634 , 0.164178, 0.26326 , 0.224457],
              [0.170778, 0.295544, 0.206207, 0.219074],
              [0.430994, 0.248644, 0.266511, 0.333039],
              [0.415324, 0.1995  , 0.241115, 0.288855]]])

    • gamma
      (chain, draw, strain)
      float64
      0.532 0.5814 0.4348 ... 0.6366 0.58

      array([[[0.532023, 0.58135 , 0.434843, 0.534044],
              [0.602685, 0.539581, 0.733511, 0.585058],
              [0.630451, 0.597629, 0.814514, 0.649502],
              [0.7366  , 0.717031, 0.505084, 0.639294],
              [0.632553, 0.933304, 0.673636, 0.597804],
              [0.483149, 0.358765, 0.485342, 0.6165  ],
              [0.618613, 0.769074, 0.663811, 0.466551],
              [0.568839, 0.434736, 0.480492, 0.671662],
              [0.526601, 0.637461, 0.612775, 0.45783 ],
              [0.664213, 0.54074 , 0.583631, 0.726396],
              [0.581525, 0.888755, 0.660172, 0.588386],
              [0.464121, 0.445068, 0.541725, 0.517733],
              [0.729252, 0.526135, 0.84669 , 0.530829],
              [0.583238, 0.582965, 0.714389, 0.615894],
              [0.6204  , 0.597953, 0.528097, 0.612643],
              [0.578326, 0.557583, 0.621011, 0.586604],
              [0.526328, 0.542489, 0.538509, 0.500688],
              [0.652097, 0.557283, 0.609697, 0.59424 ],
              [0.678017, 0.554931, 0.624087, 0.702349],
              [0.660177, 0.669473, 0.701105, 0.920669],
      ...
              [0.693976, 0.529962, 0.743694, 0.546379],
              [0.489458, 0.652452, 0.457106, 0.630765],
              [0.688274, 0.573681, 0.776939, 0.559731],
              [0.575141, 0.666426, 0.525355, 0.636601],
              [0.543359, 0.53633 , 0.653191, 0.638603],
              [0.487505, 0.596189, 0.422985, 0.470852],
              [0.716412, 0.669939, 0.733823, 0.701896],
              [0.460916, 0.582494, 0.491449, 0.581498],
              [0.519742, 0.634032, 0.57018 , 0.616262],
              [0.47168 , 0.55185 , 0.469282, 0.472171],
              [0.500134, 0.513697, 0.413252, 0.543735],
              [0.565591, 0.487246, 0.703837, 0.624566],
              [0.670913, 0.745054, 0.51993 , 0.557249],
              [0.671262, 0.571204, 0.674301, 0.673797],
              [0.584139, 0.662809, 0.587678, 0.529977],
              [0.628353, 0.672018, 0.640631, 0.71367 ],
              [0.607865, 0.505435, 0.571493, 0.397577],
              [0.684439, 0.490088, 0.642929, 0.566745],
              [0.618713, 0.823935, 0.70351 , 0.632054],
              [0.554191, 0.765503, 0.636551, 0.580032]]])

    • abundance
      (chain, draw, tube, timepoint, species)
      float64
      0.1348 1.156 ... 0.9512 0.6621

      array([[[[[0.134784 , 1.15619  ],
                [0.153593 , 1.14618  ],
                [0.174983 , 1.1348   ],
                ...,
                [1.13405  , 0.624557 ],
                [1.26509  , 0.554842 ],
                [1.40442  , 0.480713 ]],
      
               [[0.113946 , 1.08756  ],
                [0.124983 , 1.08114  ],
                [0.137066 , 1.07412  ],
                ...,
                [0.526404 , 0.847777 ],
                [0.573123 , 0.820617 ],
                [0.623351 , 0.791417 ]],
      
               [[0.131905 , 1.19961  ],
                [0.161819 , 1.1866   ],
                [0.198413 , 1.17069  ],
                ...,
      ...
                ...,
                [0.483071 , 0.748228 ],
                [0.520997 , 0.719195 ],
                [0.561517 , 0.688176 ]],
      
               [[0.112192 , 1.30229  ],
                [0.121158 , 1.29658  ],
                [0.130832 , 1.29042  ],
                ...,
                [0.409459 , 1.11306  ],
                [0.441257 , 1.09282  ],
                [0.475402 , 1.07108  ]],
      
               [[0.149399 , 1.12714  ],
                [0.165314 , 1.11791  ],
                [0.182892 , 1.10772  ],
                ...,
                [0.793853 , 0.75334  ],
                [0.869692 , 0.709351 ],
                [0.95117  , 0.662091 ]]]]])

  • Indexes: (10)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • ln_mu_max_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_mu_max_z_dim_0'))

    • ln_ks_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_ks_z_dim_0'))

    • ln_gamma_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_gamma_z_dim_0'))

    • tube
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], dtype='int64', name='tube'))

    • species
      PandasIndex

      PandasIndex(Index([1, 2], dtype='int64', name='species'))

    • sigma_y_dim_0
      PandasIndex

      PandasIndex(Index([0, 1], dtype='int64', name='sigma_y_dim_0'))

    • strain
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4], dtype='int64', name='strain'))

    • timepoint
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], dtype='int64', name='timepoint'))

  • Attributes: (4)

    created_at :

    2024-04-29T15:04:36.501119

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• posterior_predictive

  <xarray.Dataset> Size: 130kB
  Dimensions:      (chain: 1, draw: 100, measurement: 160)
  Coordinates:
    * chain        (chain) int64 8B 0
    * draw         (draw) int64 800B 0 1 2 3 4 5 6 7 8 ... 92 93 94 95 96 97 98 99
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      yrep         (chain, draw, measurement) float64 128kB 0.2038 ... 0.7842
  Attributes:
      created_at:                 2024-04-29T15:04:36.507111
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • yrep
      (chain, draw, measurement)
      float64
      0.2038 0.9137 ... 0.8199 0.7842

      array([[[0.203828, 0.91375 , 0.339964, ..., 0.79039 , 0.791635,
               0.886692],
              [0.198895, 1.44417 , 0.294053, ..., 1.00925 , 0.380898,
               0.958897],
              [0.142389, 1.12019 , 0.169761, ..., 1.21986 , 0.518106,
               1.14279 ],
              ...,
              [0.194605, 1.0105  , 0.285993, ..., 0.909459, 0.808322,
               0.70578 ],
              [0.191381, 1.3422  , 0.197284, ..., 1.03636 , 0.685514,
               1.0713  ],
              [0.185108, 1.29557 , 0.153275, ..., 1.032   , 0.819908,
               0.784214]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-29T15:04:36.507111

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• log_likelihood

  <xarray.Dataset> Size: 130kB
  Dimensions:      (chain: 1, draw: 100, measurement: 160)
  Coordinates:
    * chain        (chain) int64 8B 0
    * draw         (draw) int64 800B 0 1 2 3 4 5 6 7 8 ... 92 93 94 95 96 97 98 99
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      llik         (chain, draw, measurement) float64 128kB 2.611 ... -26.84
  Attributes:
      created_at:                 2024-04-29T15:04:36.508049
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 100
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 6 ... 94 95 96 97 98 99

      array([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • llik
      (chain, draw, measurement)
      float64
      2.611 -0.02446 ... -34.66 -26.84

      array([[[ 2.61130e+00, -2.44617e-02,  1.23532e+00, ..., -1.27257e+01,
               -6.89307e+00, -1.26069e+01],
              [ 2.42251e+00, -1.06059e+00,  1.79542e+00, ..., -7.15877e+01,
               -1.44965e+02, -9.32755e+01],
              [-2.80159e+00, -2.99513e+00, -3.61844e+00, ..., -4.53237e+01,
               -6.65011e+01, -5.60089e+01],
              ...,
              [ 2.96624e+00, -2.88618e+00,  8.19052e-01, ..., -3.03733e+01,
               -2.65193e+01, -3.62879e+01],
              [ 1.41679e+00, -5.78144e-01, -8.87229e-01, ..., -3.19253e+01,
               -3.09168e+01, -3.60352e+01],
              [-2.47380e-01,  8.73777e-01, -5.73510e+00, ..., -2.20712e+01,
               -3.46581e+01, -2.68351e+01]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([ 0,  1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17,
             18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35,
             36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53,
             54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71,
             72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89,
             90, 91, 92, 93, 94, 95, 96, 97, 98, 99],
            dtype='int64', name='draw'))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-29T15:04:36.508049

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• sample_stats

  <xarray.Dataset> Size: 11kB
  Dimensions:          (chain: 1, draw: 200)
  Coordinates:
    * chain            (chain) int64 8B 0
    * draw             (draw) int64 2kB 0 1 2 3 4 5 6 ... 194 195 196 197 198 199
  Data variables:
      lp               (chain, draw) float64 2kB -49.78 -49.78 ... -60.39 -61.47
      acceptance_rate  (chain, draw) float64 2kB 0.9419 0.0 0.0 ... 0.6652 0.674
      step_size        (chain, draw) float64 2kB 0.03125 12.94 ... 0.06485 0.06485
      tree_depth       (chain, draw) int64 2kB 7 0 0 2 8 7 6 6 ... 6 6 6 6 6 6 6 3
      n_steps          (chain, draw) int64 2kB 127 1 1 6 255 255 ... 63 63 63 63 7
      diverging        (chain, draw) bool 200B False True True ... False False
      energy           (chain, draw) float64 2kB 77.45 73.93 71.99 ... 81.05 89.4
  Attributes:
      created_at:                 2024-04-29T15:04:36.505485
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 1
    • draw: 200
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0

      array([0])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 195 196 197 198 199

      array([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,
              14,  15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,
              28,  29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,
              42,  43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,
              56,  57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,
              70,  71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,
              84,  85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,
              98,  99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111,
             112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125,
             126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139,
             140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153,
             154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167,
             168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181,
             182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195,
             196, 197, 198, 199])

  • Data variables: (7)
    • lp
      (chain, draw)
      float64
      -49.78 -49.78 ... -60.39 -61.47

      array([[-49.7805, -49.7805, -49.7805, -49.7805, -51.5668, -58.4996,
              -56.3268, -64.6768, -56.8653, -59.6851, -60.7597, -62.1438,
              -58.7905, -58.2967, -54.2829, -64.3672, -58.514 , -59.2999,
              -62.7017, -62.8461, -58.5   , -54.3902, -55.1173, -58.5765,
              -60.6034, -60.6479, -59.0477, -60.4051, -56.6057, -52.012 ,
              -54.8291, -52.8148, -48.9321, -51.0286, -52.0858, -65.3476,
              -66.7186, -62.0462, -63.3358, -64.4222, -58.7215, -55.801 ,
              -57.0435, -59.6045, -63.2893, -55.6438, -58.3681, -80.4703,
              -65.497 , -64.6605, -65.504 , -64.381 , -75.6417, -55.1675,
              -58.3064, -61.3141, -60.9572, -55.8952, -64.1739, -69.1418,
              -65.3524, -63.9233, -65.042 , -67.5354, -63.4723, -64.7666,
              -60.8848, -61.1282, -75.2364, -66.2174, -71.6069, -59.4215,
              -60.3784, -66.1273, -64.4596, -55.4351, -55.7801, -55.9407,
              -49.7089, -48.98  , -57.7659, -56.5001, -58.6635, -63.3792,
              -52.0382, -59.8635, -58.8561, -58.9825, -59.529 , -56.9501,
              -57.3132, -53.691 , -55.7869, -54.4031, -52.3796, -65.4703,
              -66.2117, -68.9405, -70.9659, -69.2255, -57.2979, -60.7441,
              -64.1444, -55.7779, -58.4294, -57.4225, -63.5947, -65.2081,
              -56.8927, -64.2135, -66.056 , -59.6172, -63.073 , -59.8313,
              -60.9658, -53.3057, -52.438 , -51.4617, -51.5673, -58.7971,
              -55.5788, -56.7901, -57.4201, -56.9684, -64.0354, -66.2677,
              -61.7488, -61.4998, -68.0562, -56.8178, -53.7206, -53.613 ,
              -55.6649, -51.3548, -63.5595, -62.6105, -62.1237, -58.6289,
              -56.0824, -63.5739, -65.9644, -62.043 , -58.6055, -65.8475,
              -55.0829, -55.1142, -58.28  , -69.1147, -61.0752, -61.0459,
              -66.8474, -55.4503, -57.5563, -61.2264, -63.6987, -61.7274,
              -59.6398, -61.8955, -59.0384, -62.0704, -68.1986, -62.5255,
              -59.8245, -57.9484, -54.1836, -54.6873, -56.6693, -54.2953,
              -48.8965, -59.1588, -61.323 , -67.6306, -65.9698, -63.783 ,
              -61.5224, -56.9751, -61.5855, -59.0472, -58.8017, -68.1409,
              -63.4529, -65.2902, -62.5962, -58.4732, -59.1248, -61.3362,
              -58.0509, -58.6503, -57.8498, -57.7526, -58.3588, -56.1637,
              -59.3344, -62.0214, -57.9878, -50.348 , -58.7765, -53.1134,
              -60.3864, -61.4669]])

    • acceptance_rate
      (chain, draw)
      float64
      0.9419 0.0 0.0 ... 0.6652 0.674

      array([[9.41945e-01, 0.00000e+00, 0.00000e+00, 5.43029e-27, 9.69413e-01,
              9.71088e-01, 9.81677e-01, 8.66360e-01, 1.00000e+00, 5.93031e-01,
              9.95326e-01, 4.28541e-01, 9.99667e-01, 8.63743e-01, 7.50527e-01,
              5.63470e-01, 9.94028e-01, 8.33743e-01, 8.40846e-01, 9.61597e-01,
              2.66597e-01, 9.79747e-01, 7.98756e-01, 8.30952e-01, 8.89403e-01,
              9.65529e-01, 1.44779e-01, 9.95206e-01, 9.99497e-01, 5.72402e-01,
              8.77060e-01, 9.22804e-01, 6.95844e-01, 8.87962e-01, 4.05398e-01,
              8.17986e-01, 9.95249e-01, 9.68231e-01, 9.48038e-01, 3.68988e-01,
              9.87265e-01, 5.78946e-01, 8.61113e-01, 5.79879e-01, 9.48072e-01,
              9.30252e-01, 6.82385e-01, 5.83503e-01, 9.48999e-01, 9.98699e-01,
              9.02747e-01, 5.43975e-01, 7.82631e-01, 1.00000e+00, 3.64436e-01,
              9.51437e-01, 9.58085e-01, 9.64601e-01, 2.65468e-01, 8.81167e-01,
              9.99065e-01, 7.22565e-01, 6.98196e-01, 8.63158e-01, 9.90163e-01,
              6.58810e-01, 7.08150e-01, 9.45426e-01, 7.15708e-01, 9.18037e-01,
              7.05524e-01, 8.95741e-01, 6.02802e-01, 8.59617e-01, 4.85879e-01,
              9.97588e-01, 9.74425e-01, 2.29309e-01, 9.98658e-01, 7.69316e-01,
              7.80717e-01, 9.95105e-01, 5.30601e-01, 8.99861e-01, 9.99996e-01,
              4.09616e-01, 9.60829e-01, 8.70729e-01, 9.96985e-01, 5.11469e-01,
              9.95059e-01, 7.95041e-01, 3.27112e-01, 9.80104e-01, 7.80299e-01,
              6.48863e-01, 9.30339e-01, 9.13457e-01, 9.02697e-01, 9.98233e-01,
              9.92337e-01, 7.60500e-01, 7.97596e-01, 7.82566e-01, 5.58097e-01,
              9.12863e-01, 6.97505e-01, 8.69133e-01, 8.52626e-01, 7.08377e-01,
              9.77767e-01, 8.88363e-01, 9.71751e-01, 9.08548e-01, 9.37000e-01,
              7.79121e-01, 8.61566e-01, 7.08951e-01, 5.76561e-01, 7.84213e-01,
              8.31022e-01, 6.09673e-01, 7.28554e-01, 5.61963e-01, 7.31386e-01,
              9.90962e-01, 9.97421e-01, 9.74054e-01, 7.88956e-01, 9.95493e-01,
              9.96285e-01, 5.46640e-01, 7.99156e-01, 8.36111e-01, 6.48933e-01,
              8.65554e-01, 7.80286e-01, 9.98339e-01, 7.98633e-01, 8.09184e-01,
              9.96272e-01, 7.31876e-01, 9.36707e-01, 7.99823e-01, 9.49830e-01,
              7.04399e-01, 6.62776e-01, 5.35788e-01, 9.39701e-01, 9.68509e-01,
              6.13213e-01, 9.65918e-01, 8.26226e-01, 6.58482e-01, 8.93352e-01,
              8.44683e-01, 9.78547e-01, 9.29284e-01, 9.98874e-01, 9.17985e-01,
              8.90315e-01, 9.99941e-01, 8.06005e-01, 9.61584e-01, 9.89137e-01,
              5.56961e-01, 9.42879e-01, 9.14851e-01, 7.78335e-01, 6.23410e-01,
              9.58390e-01, 7.50044e-01, 1.00000e+00, 8.65638e-01, 9.99162e-01,
              9.14732e-01, 8.28930e-01, 9.98934e-01, 7.64093e-01, 5.54538e-01,
              6.67803e-01, 8.18442e-01, 6.21054e-01, 9.90968e-01, 8.23509e-01,
              9.69964e-01, 9.68896e-01, 8.70260e-01, 5.97765e-01, 7.36700e-01,
              6.20001e-01, 9.99838e-01, 8.62094e-01, 6.78376e-01, 9.92441e-01,
              9.77883e-01, 6.60522e-01, 8.15210e-01, 6.65203e-01, 6.74009e-01]])

    • step_size
      (chain, draw)
      float64
      0.03125 12.94 ... 0.06485 0.06485

      array([[ 0.03125  , 12.9444   ,  2.12025  ,  0.205423 ,  0.0157802,
               0.0197512,  0.0281925,  0.0450227,  0.0526442,  0.0965212,
               0.0498406,  0.0929272,  0.0292566,  0.0561432,  0.0704648,
               0.0623693,  0.0312014,  0.0588948,  0.0678869,  0.0798937,
               0.134584 ,  0.0288237,  0.0512938,  0.053454 ,  0.0611101,
               0.0824052,  0.137203 ,  0.0225909,  0.0409631,  0.0744223,
               0.0415661,  0.0536181,  0.0779063,  0.06135  ,  0.0806919,
               0.0297072,  0.0325918,  0.056485 ,  0.0905481,  0.136865 ,
               0.0475062,  0.079109 ,  0.0472458,  0.0571366,  0.0345256,
               0.0515771,  0.073367 ,  0.0572012,  0.0353283,  0.0522696,
               0.0864759,  0.113503 ,  0.0644259,  0.0639396,  0.104529 ,
               0.0398181,  0.0581024,  0.0856474,  0.127482 ,  0.0398133,
               0.0492451,  0.0787384,  0.0683778,  0.0563997,  0.0665939,
               0.10311  ,  0.0782211,  0.066097 ,  0.0923997,  0.0793822,
               0.104286 ,  0.0877799,  0.109717 ,  0.0748453,  0.0867288,
               0.0468504,  0.0718579,  0.10472  ,  0.03407  ,  0.0520992,
               0.0502884,  0.049659 ,  0.0747858,  0.045151 ,  0.0562669,
               0.0849785,  0.0408606,  0.0570892,  0.0669157,  0.0996189,
               0.0588169,  0.0869709,  0.0879721,  0.0370423,  0.0530887,
               0.0523396,  0.0404713,  0.0526503,  0.0662557,  0.0815929,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502,
               0.0648502,  0.0648502,  0.0648502,  0.0648502,  0.0648502]])

    • tree_depth
      (chain, draw)
      int64
      7 0 0 2 8 7 6 6 ... 6 6 6 6 6 6 6 3

      array([[ 7,  0,  0,  2,  8,  7,  6,  6,  3,  5, 10,  3,  5,  5,  6,  5,
               7,  6,  6,  6,  5,  7,  6,  4,  6,  6,  5,  7,  7,  5,  7,  3,
               4,  4,  4,  4,  5,  5,  5,  5,  6,  6,  4,  6,  4,  6,  5,  6,
               7,  4,  6,  5,  6,  3,  5,  7,  6,  5,  5,  7,  6,  5,  6,  3,
               6,  6,  6,  6,  5,  6,  5,  5,  5,  5,  6,  3,  5,  5,  7,  4,
               8,  6,  6,  3,  6,  6,  7,  6,  6,  6,  6,  6,  5,  7,  6,  6,
               7,  6,  6,  6,  5,  6,  3,  6,  6,  6,  6,  6,  6,  6,  6,  6,
               6,  6,  6,  6,  4,  6,  5,  6,  6,  6,  6,  6,  6,  3,  5,  6,
               6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,
               6,  6,  6,  6,  5,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,
               6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,  6,
               6,  6,  6,  3,  3,  6,  6,  6,  6,  6,  6,  6,  3,  5,  4,  6,
               6,  6,  6,  6,  6,  6,  6,  3]])

    • n_steps
      (chain, draw)
      int64
      127 1 1 6 255 255 ... 63 63 63 63 7

      array([[ 127,    1,    1,    6,  255,  255,   95,   79,    7,   31, 1023,
                 7,   63,   63,   63,   39,  127,   63,   63,   63,   31,  127,
               127,   31,   63,   63,   31,  127,  127,   63,  127,    7,   23,
                31,   31,   15,   63,   39,   63,   31,   63,   63,   15,   63,
                15,   63,   31,   63,  127,   31,   63,   31,   63,    7,   31,
               127,   63,   31,   31,  127,   63,   31,   63,    7,   63,   63,
                63,   63,   63,   63,   31,   31,   31,   31,   63,   15,   63,
                31,  127,   31,  319,   63,   63,    7,   63,   63,  127,   63,
                63,  127,   63,   63,   63,  127,   63,  127,  127,   63,   63,
                63,   47,   63,   15,   63,   63,   63,   63,   63,   63,   63,
                63,   63,   63,   63,   63,   63,   23,   63,   39,   63,   63,
                63,   63,   63,   63,   15,   39,   63,   63,   63,   63,   63,
                63,   63,   63,   63,   63,   63,   63,   63,   63,   63,   63,
                63,   63,   63,   63,   63,   63,   63,   63,   63,   63,   63,
                63,   63,   63,   63,   63,   63,   63,   63,   63,   63,   63,
                63,   63,   63,   63,   63,   63,   63,   63,   63,   63,   63,
                63,   63,   63,    7,    7,   63,   63,   63,   63,   63,   63,
                63,    7,   39,   31,   63,   63,   63,   63,   63,   63,   63,
                63,    7]])

    • diverging
      (chain, draw)
      bool
      False True True ... False False

      array([[False,  True,  True,  True, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False, False, False, False, False, False, False, False,
              False, False]])

    • energy
      (chain, draw)
      float64
      77.45 73.93 71.99 ... 81.05 89.4

      array([[ 77.4463,  73.9347,  71.987 ,  71.955 ,  73.9218,  74.3722,
               85.0013,  85.5619,  84.369 ,  82.7776,  83.1049,  85.0976,
               84.9294,  87.5884,  87.2614,  93.5409,  84.5439,  80.4855,
               81.6282,  82.3561,  87.0736,  79.087 ,  79.029 ,  87.9453,
               80.1899,  81.8837,  88.0438,  86.6437,  82.2471,  85.3187,
               78.8758,  73.0634,  74.166 ,  70.811 ,  85.9564,  85.6963,
               92.6561,  94.81  ,  89.4784,  91.4095,  89.1848,  85.5199,
               79.0862,  95.3734,  82.0398,  86.0642,  83.2618, 100.795 ,
              113.495 ,  92.9701,  89.1584,  94.0134, 100.88  ,  88.132 ,
               82.5482,  85.2362,  87.1318,  78.3482,  85.287 ,  99.9056,
               95.3312,  89.9039,  96.5342,  93.3195,  92.5165,  87.7339,
               94.0473,  85.0565,  95.8899, 100.32  ,  97.7212,  92.0177,
               83.7795,  90.6626,  99.9094,  80.9606,  71.7648,  87.358 ,
               74.393 ,  76.5634,  72.4908,  74.4653,  84.7449,  79.1042,
               79.8241,  83.4537,  83.5795,  84.0898,  76.2124,  84.7643,
               79.6166,  75.7789,  83.3553,  76.4545,  81.5055,  87.2923,
               98.3156,  96.9818,  98.8975,  95.8657,  89.1768,  80.1892,
               90.0923,  93.3981,  88.3559,  79.3816,  84.8027,  92.956 ,
               85.8329,  83.8131,  91.878 ,  92.7243,  86.7221,  88.5955,
               85.6743,  85.6615,  75.5816,  75.9325,  79.1071,  77.9917,
               81.0492,  85.8785,  81.0481,  89.1259,  84.8208,  86.3186,
               84.3283,  84.2514,  93.0195,  89.9554,  77.6913,  82.3784,
               77.5864,  75.4855,  79.0557,  89.3749,  92.4028,  84.5889,
               82.0349,  85.8493,  93.4445,  98.48  ,  82.0963,  86.4074,
               86.5297,  82.5834,  82.482 ,  97.8201,  92.0954,  81.6402,
              102.51  ,  93.4911,  82.7224,  84.4392,  91.3629,  91.8881,
               85.5848,  84.0381,  83.9987,  85.1426,  90.9245,  90.6896,
               90.3969,  82.3485,  76.68  ,  84.1584,  74.063 ,  76.4153,
               74.21  ,  75.1773,  83.7979,  97.4695,  85.8742,  95.1545,
               86.9542,  84.3535,  86.2992,  84.4119,  87.2793,  87.533 ,
               97.7188,  93.0237, 102.14  ,  86.7154,  85.1507,  82.2143,
               83.9534,  81.9174,  85.0608,  86.7636,  91.7619,  77.2411,
               79.7312,  87.0638,  84.7309,  72.9301,  78.161 ,  80.363 ,
               81.045 ,  89.3986]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             190, 191, 192, 193, 194, 195, 196, 197, 198, 199],
            dtype='int64', name='draw', length=200))

  • Attributes: (4)

    created_at :

    2024-04-29T15:04:36.505485

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  

We can find the prior intervals for the true abundance and plot them in the graph.

prior_abundances = idata_prior.posterior["abundance"]

n_sample = 20
chains = rng.choice(prior_abundances.coords["chain"].values, n_sample)
draws = rng.choice(prior_abundances.coords["draw"].values, n_sample)
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

for ax, species_i in zip(axes, species):
    for tube_j in tubes:
        for chain, draw in zip(chains, draws):
            timeseries = prior_abundances.sel(chain=chain, draw=draw, tube=tube_j, species=species_i)
            ax.plot(
                timepoints.values, 
                timeseries.values,
                alpha=0.5, color="skyblue", zorder=-1
            )
f.savefig("img/monod_priors.png")

Sample in posterior mode

stan_input_posterior = stan_input_common | {
    "y": fake_measurements["simulated_measurement"],
    "likelihood": 1
}
mcmc_posterior = model.sample(
    data=stan_input_posterior,
    iter_warmup=300,
    iter_sampling=300,
    chains=4,
    refresh=1,
    inits=true_param_values,
    seed=SEED,
)
idata_posterior = az.from_cmdstanpy(
    mcmc_posterior,
    dims=dims,
    coords=coords,
    posterior_predictive={"y": "yrep"},
    log_likelihood="llik"
)
idata_posterior

17:04:37 - cmdstanpy - INFO - CmdStan start processing
17:10:39 - cmdstanpy - INFO - CmdStan done processing.
17:10:39 - cmdstanpy - WARNING - Non-fatal error during sampling:
Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: ode_bdf_tol: initial state[1] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: ode_bdf_tol: initial state[1] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: CVode(cvodes_mem, t_final, nv_state_, &t_init, CV_NORMAL) failed with error flag -4: 
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: ode_bdf_tol: initial state[1] is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: ode_bdf_tol: ode parameters and data is inf, but must be finite! (in 'monod.stan', line 56, column 4 to line 60, column 79)
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
    Exception: lognormal_lpdf: Location parameter is nan, but must be finite! (in 'monod.stan', line 85, column 6 to column 64)
Consider re-running with show_console=True if the above output is unclear!

arviz.InferenceData

• posterior

  <xarray.Dataset> Size: 7MB
  Dimensions:            (chain: 4, draw: 300, ln_mu_max_z_dim_0: 4,
                          ln_ks_z_dim_0: 4, ln_gamma_z_dim_0: 4, tube: 16,
                          species: 2, sigma_y_dim_0: 2, strain: 4, timepoint: 20)
  Coordinates:
    * chain              (chain) int64 32B 0 1 2 3
    * draw               (draw) int64 2kB 0 1 2 3 4 5 ... 294 295 296 297 298 299
    * ln_mu_max_z_dim_0  (ln_mu_max_z_dim_0) int64 32B 0 1 2 3
    * ln_ks_z_dim_0      (ln_ks_z_dim_0) int64 32B 0 1 2 3
    * ln_gamma_z_dim_0   (ln_gamma_z_dim_0) int64 32B 0 1 2 3
    * tube               (tube) int64 128B 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    * species            (species) int64 16B 1 2
    * sigma_y_dim_0      (sigma_y_dim_0) int64 16B 0 1
    * strain             (strain) int64 32B 1 2 3 4
    * timepoint          (timepoint) int64 160B 1 2 3 4 5 6 ... 15 16 17 18 19 20
  Data variables: (12/15)
      ln_mu_max_z        (chain, draw, ln_mu_max_z_dim_0) float64 38kB -0.986 ....
      ln_ks_z            (chain, draw, ln_ks_z_dim_0) float64 38kB -0.3128 ... ...
      ln_gamma_z         (chain, draw, ln_gamma_z_dim_0) float64 38kB 1.868 ......
      a_mu_max           (chain, draw) float64 10kB -1.538 -1.456 ... -1.439
      a_ks               (chain, draw) float64 10kB -1.294 -1.327 ... -1.217
      a_gamma            (chain, draw) float64 10kB -0.6778 -0.6196 ... -0.5872
      ...                 ...
      species_zero       (chain, draw, tube, species) float64 307kB 0.1234 ... ...
      sigma_y            (chain, draw, sigma_y_dim_0) float64 19kB 0.08303 ... ...
      mu_max             (chain, draw, strain) float64 38kB 0.1683 ... 0.239
      ks                 (chain, draw, strain) float64 38kB 0.2512 ... 0.2017
      gamma              (chain, draw, strain) float64 38kB 0.6667 ... 0.5471
      abundance          (chain, draw, tube, timepoint, species) float64 6MB 0....
  Attributes:
      created_at:                 2024-04-29T15:10:39.435568
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • ln_mu_max_z_dim_0: 4
    • ln_ks_z_dim_0: 4
    • ln_gamma_z_dim_0: 4
    • tube: 16
    • species: 2
    • sigma_y_dim_0: 2
    • strain: 4
    • timepoint: 20
  • Coordinates: (10)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • ln_mu_max_z_dim_0
      (ln_mu_max_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_ks_z_dim_0
      (ln_ks_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • ln_gamma_z_dim_0
      (ln_gamma_z_dim_0)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • tube
      (tube)
      int64
      1 2 3 4 5 6 7 ... 11 12 13 14 15 16

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16])

    • species
      (species)
      int64
      1 2

      array([1, 2])

    • sigma_y_dim_0
      (sigma_y_dim_0)
      int64
      0 1

      array([0, 1])

    • strain
      (strain)
      int64
      1 2 3 4

      array([1, 2, 3, 4])

    • timepoint
      (timepoint)
      int64
      1 2 3 4 5 6 7 ... 15 16 17 18 19 20

      array([ 1,  2,  3,  4,  5,  6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18,
             19, 20])

  • Data variables: (15)
    • ln_mu_max_z
      (chain, draw, ln_mu_max_z_dim_0)
      float64
      -0.986 0.1325 ... 0.1963 0.03514

      array([[[-0.985973  ,  0.132507  ,  0.673448  ,  0.239702  ],
              [-0.978587  , -0.0403766 , -0.0395584 , -0.221628  ],
              [-0.859079  ,  0.0134602 ,  0.441481  ,  0.438556  ],
              ...,
              [-1.01057   ,  0.15906   ,  0.567065  ,  0.950912  ],
              [-0.347702  ,  0.68927   ,  1.29645   ,  1.57775   ],
              [-0.957113  ,  0.0788746 ,  0.530912  ,  0.306463  ]],
      
             [[-0.761376  ,  0.967731  ,  0.968887  ,  0.487544  ],
              [-0.663488  ,  1.15833   ,  1.27743   ,  0.737162  ],
              [-1.38255   , -0.324846  ,  0.105306  ,  1.4279    ],
              ...,
              [-0.763366  ,  0.522852  ,  1.03871   ,  0.717876  ],
              [-1.00791   ,  0.0510227 ,  0.428909  ,  0.612282  ],
              [-0.99481   ,  0.597276  ,  0.709687  ,  0.620925  ]],
      
             [[-1.19974   , -0.0359503 ,  0.478918  ,  1.36692   ],
              [-0.967367  ,  0.142563  ,  0.486198  ,  1.15366   ],
              [-0.122563  ,  1.02532   ,  1.8542    ,  1.7221    ],
              ...,
              [ 0.895368  ,  1.58192   ,  1.61797   ,  1.91926   ],
              [-1.10702   ,  0.378512  ,  1.13657   ,  1.48149   ],
              [-0.0435118 ,  1.00607   ,  1.26552   ,  1.41795   ]],
      
             [[-1.52946   ,  0.188188  ,  0.872712  , -0.00323341],
              [-0.612154  ,  0.479953  ,  0.333622  ,  0.964259  ],
              [-0.370222  ,  0.595919  ,  1.19821   ,  0.807859  ],
              ...,
              [-0.86944   ,  0.292368  ,  0.280778  ,  0.594227  ],
              [-0.594137  ,  0.259787  ,  0.299896  ,  0.817156  ],
              [-2.01333   , -0.719191  ,  0.196339  ,  0.0351351 ]]])

    • ln_ks_z
      (chain, draw, ln_ks_z_dim_0)
      float64
      -0.3128 0.1205 ... -0.5083 -1.078

      array([[[-0.312792 ,  0.120541 ,  0.936168 , -1.6219   ],
              [ 0.600648 ,  1.43247  , -0.902656 , -2.78253  ],
              [-0.0155619, -0.125625 , -0.501553 , -0.937056 ],
              ...,
              [-0.894164 , -0.828557 , -0.928359 ,  0.303595 ],
              [-0.412389 , -1.776    , -0.740069 ,  0.411631 ],
              [ 1.24878  ,  1.31362  ,  1.08855  ,  0.369811 ]],
      
             [[ 0.159619 ,  2.15272  ,  0.886759 , -1.93281  ],
              [-0.393275 ,  1.64963  ,  0.778617 , -1.28771  ],
              [-0.31039  , -1.0705   , -1.23761  ,  2.99249  ],
              ...,
              [-0.239405 ,  0.628227 ,  0.497207 , -0.303942 ],
              [-0.146671 , -0.579092 , -0.138931 , -0.484474 ],
              [-0.903065 ,  1.18116  ,  0.245618 , -0.512992 ]],
      
             [[ 0.287773 , -0.207406 ,  0.219969 ,  1.91013  ],
              [-0.553512 ,  0.0345259, -0.191872 ,  1.39211  ],
              [-0.503636 , -0.705214 ,  0.751475 , -0.61968  ],
              ...,
              [ 1.35935  , -0.42849  , -1.98763  , -1.87782  ],
              [-2.31747  ,  0.0278805,  1.3997   ,  2.358    ],
              [ 0.0346256,  0.430672 , -0.493994 , -0.338745 ]],
      
             [[ 0.995386 ,  1.60561  ,  2.14047  , -0.0385813],
              [ 0.143883 ,  0.283359 , -1.69068  , -0.231424 ],
              [-0.0465751, -1.31214  , -0.0985126, -2.76062  ],
              ...,
              [ 0.913904 ,  1.4992   ,  0.0274485,  0.325106 ],
              [ 1.67616  ,  1.05239  , -0.279328 ,  0.599958 ],
              [-1.86981  , -1.50896  , -0.508265 , -1.078    ]]])

    • ln_gamma_z
      (chain, draw, ln_gamma_z_dim_0)
      float64
      1.868 -0.1992 2.092 ... 1.19 -0.126

      array([[[ 1.86775  , -0.199213 ,  2.09216  ,  0.969051 ],
              [ 1.11175  , -0.432373 ,  1.80173  ,  0.272905 ],
              [ 1.31896  , -0.779982 ,  0.907051 ,  0.438461 ],
              ...,
              [ 2.22018  , -0.267309 ,  0.574084 , -0.287221 ],
              [ 0.861789 , -0.745576 ,  0.232874 ,  0.888324 ],
              [ 0.539166 , -1.31896  ,  0.838447 , -0.0970454]],
      
             [[-0.646875 , -1.29752  ,  0.456219 , -1.26381  ],
              [-1.1582   , -1.71352  ,  0.324232 , -0.752678 ],
              [ 0.740235 , -0.988123 ,  0.607942 ,  0.458298 ],
              ...,
              [-0.460292 , -0.869187 , -0.430669 , -0.976219 ],
              [-0.844501 , -2.33048  , -0.146924 , -1.13001  ],
              [ 0.0255257, -1.02759  , -0.524137 , -0.99071  ]],
      
             [[ 1.76129  , -1.63949  ,  0.923725 ,  0.615935 ],
              [ 0.067796 , -1.58883  ,  0.203878 , -1.24254  ],
              [-1.69953  , -1.98448  ,  1.01397  , -0.961183 ],
              ...,
              [-0.0901274, -2.22447  ,  0.193064 , -1.20649  ],
              [ 0.387707 , -1.1007   ,  1.08663  ,  0.292302 ],
              [ 1.21702  , -1.03814  ,  1.08974  ,  0.463001 ]],
      
             [[-1.41498  , -2.37936  ,  0.961229 , -0.837624 ],
              [ 1.67642  , -1.50747  ,  1.18843  , -0.341188 ],
              [ 1.0435   , -2.19982  ,  1.19638  ,  0.239922 ],
              ...,
              [ 0.212532 , -1.28613  ,  0.49534  , -0.276068 ],
              [ 1.17352  , -1.63135  ,  0.454401 , -0.580538 ],
              [ 0.935913 , -2.29589  ,  1.18997  , -0.12595  ]]])

    • a_mu_max
      (chain, draw)
      float64
      -1.538 -1.456 ... -1.543 -1.439

      array([[-1.53848, -1.45583, -1.54529, ..., -1.61887, -1.72982, -1.46351],
             [-1.61545, -1.65676, -1.47113, ..., -1.57562, -1.54436, -1.56511],
             [-1.48673, -1.56867, -1.77797, ..., -1.91933, -1.58584, -1.73161],
             [-1.36543, -1.59679, -1.681  , ..., -1.5106 , -1.54297, -1.439  ]])

    • a_ks
      (chain, draw)
      float64
      -1.294 -1.327 ... -1.413 -1.217

      array([[-1.29419, -1.32717, -1.27184, ..., -1.42591, -1.17353, -1.35674],
             [-1.44741, -1.29788, -1.30569, ..., -1.23928, -1.34055, -1.28647],
             [-1.26071, -1.35109, -1.33275, ..., -1.3317 , -1.28532, -1.24424],
             [-1.32257, -1.22667, -1.34155, ..., -1.37826, -1.4133 , -1.21741]])

    • a_gamma
      (chain, draw)
      float64
      -0.6778 -0.6196 ... -0.4948 -0.5872

      array([[-0.677817, -0.619644, -0.589318, ..., -0.555483, -0.583177,
              -0.448795],
             [-0.45775 , -0.439628, -0.576847, ..., -0.401179, -0.406173,
              -0.411666],
             [-0.507466, -0.383965, -0.503686, ..., -0.460933, -0.510647,
              -0.585694],
             [-0.504623, -0.536065, -0.587293, ..., -0.481801, -0.494799,
              -0.587203]])

    • t_mu_max
      (chain, draw)
      float64
      0.2469 0.3033 ... 0.2697 0.2252

      array([[0.24691 , 0.303275, 0.287962, ..., 0.260431, 0.262664, 0.249793],
             [0.260647, 0.264728, 0.240784, ..., 0.251703, 0.244429, 0.254446],
             [0.228664, 0.269656, 0.223726, ..., 0.243759, 0.252198, 0.258899],
             [0.206917, 0.25612 , 0.214311, ..., 0.273427, 0.269736, 0.225171]])

    • t_ks
      (chain, draw)
      float64
      0.2797 0.2837 ... 0.2578 0.3558

      array([[0.279737, 0.283693, 0.276972, ..., 0.315361, 0.280463, 0.31751 ],
             [0.333963, 0.358231, 0.269969, ..., 0.325036, 0.260704, 0.292198],
             [0.3348  , 0.344453, 0.304161, ..., 0.328656, 0.263126, 0.318937],
             [0.338368, 0.279188, 0.303897, ..., 0.261964, 0.257809, 0.355768]])

    • t_gamma
      (chain, draw)
      float64
      0.1459 0.1343 ... 0.1502 0.1268

      array([[0.145884, 0.134349, 0.150049, ..., 0.141701, 0.144763, 0.132735],
             [0.124359, 0.119967, 0.150477, ..., 0.143735, 0.159177, 0.152041],
             [0.143358, 0.126917, 0.158984, ..., 0.134735, 0.151274, 0.131313],
             [0.137908, 0.138309, 0.134802, ..., 0.148068, 0.150219, 0.126842]])

    • species_zero
      (chain, draw, tube, species)
      float64
      0.1234 1.159 ... 0.1416 1.039

      array([[[[0.12344  , 1.15863  ],
               [0.13965  , 1.32984  ],
               [0.141506 , 1.2288   ],
               ...,
               [0.119436 , 1.14172  ],
               [0.13743  , 1.2384   ],
               [0.145679 , 1.12817  ]],
      
              [[0.131801 , 1.36376  ],
               [0.133478 , 1.3025   ],
               [0.137145 , 1.25831  ],
               ...,
               [0.121751 , 1.27508  ],
               [0.129996 , 1.18512  ],
               [0.139906 , 1.06386  ]],
      
              [[0.133001 , 1.1553   ],
               [0.141494 , 1.33323  ],
               [0.140589 , 1.30809  ],
               ...,
      ...
               ...,
               [0.106831 , 1.20751  ],
               [0.130238 , 1.14153  ],
               [0.141446 , 1.10522  ]],
      
              [[0.123493 , 1.22402  ],
               [0.138039 , 1.2903   ],
               [0.146198 , 1.30497  ],
               ...,
               [0.107893 , 1.20567  ],
               [0.131385 , 1.17438  ],
               [0.142079 , 1.09701  ]],
      
              [[0.124253 , 1.10582  ],
               [0.138368 , 1.22654  ],
               [0.129403 , 1.29083  ],
               ...,
               [0.111359 , 1.15863  ],
               [0.121597 , 1.22754  ],
               [0.14162  , 1.039    ]]]])

    • sigma_y
      (chain, draw, sigma_y_dim_0)
      float64
      0.08303 0.1086 ... 0.09184 0.1082

      array([[[0.0830255, 0.108566 ],
              [0.0875628, 0.0966766],
              [0.0990726, 0.0931656],
              ...,
              [0.0812123, 0.11327  ],
              [0.0735859, 0.113955 ],
              [0.0920056, 0.106174 ]],
      
             [[0.0788326, 0.0986462],
              [0.0772889, 0.10314  ],
              [0.0834216, 0.108684 ],
              ...,
              [0.0831573, 0.109597 ],
              [0.0782077, 0.116434 ],
              [0.0806773, 0.103249 ]],
      
             [[0.0778783, 0.11835  ],
              [0.0750315, 0.11725  ],
              [0.0874491, 0.114861 ],
              ...,
              [0.0806572, 0.0998167],
              [0.078145 , 0.118014 ],
              [0.0794178, 0.097273 ]],
      
             [[0.0806206, 0.111619 ],
              [0.0862446, 0.103468 ],
              [0.0998071, 0.120444 ],
              ...,
              [0.0737188, 0.113346 ],
              [0.0720314, 0.116417 ],
              [0.0918368, 0.108161 ]]])

    • mu_max
      (chain, draw, strain)
      float64
      0.1683 0.2218 ... 0.2479 0.239

      array([[[0.168313, 0.221847, 0.253548, 0.227797],
              [0.173321, 0.230369, 0.230426, 0.218047],
              [0.166515, 0.214079, 0.24216 , 0.241956],
              ...,
              [0.152277, 0.206502, 0.229652, 0.253797],
              [0.16184 , 0.212508, 0.249252, 0.268366],
              [0.182211, 0.236027, 0.264242, 0.249834]],
      
             [[0.163017, 0.255838, 0.255915, 0.22574 ],
              [0.160029, 0.25921 , 0.267513, 0.231862],
              [0.164635, 0.212387, 0.235565, 0.323904],
              ...,
              [0.170715, 0.235978, 0.268697, 0.247851],
              [0.16684 , 0.216127, 0.237041, 0.247907],
              [0.162311, 0.243378, 0.25044 , 0.244847]],
      
             [[0.171862, 0.22426 , 0.25228 , 0.309077],
              [0.160489, 0.216486, 0.237506, 0.284342],
              [0.16441 , 0.21255 , 0.255857, 0.248405],
              ...,
              [0.182487, 0.215731, 0.217635, 0.23422 ],
              [0.154892, 0.225288, 0.272751, 0.297539],
              [0.175016, 0.229663, 0.24562 , 0.255507]],
      
             [[0.186021, 0.265408, 0.305793, 0.255101],
              [0.173154, 0.229039, 0.220614, 0.259287],
              [0.171985, 0.21155 , 0.240698, 0.221381],
              ...,
              [0.174065, 0.239152, 0.238395, 0.259728],
              [0.182094, 0.22926 , 0.231753, 0.266453],
              [0.150718, 0.201706, 0.247884, 0.239048]]])

    • ks
      (chain, draw, strain)
      float64
      0.2512 0.2835 ... 0.247 0.2017

      array([[[0.251155, 0.283521, 0.356184, 0.17414 ],
              [0.314501, 0.398205, 0.205308, 0.120446],
              [0.279109, 0.270729, 0.243958, 0.216237],
              ...,
              [0.181247, 0.185036, 0.179303, 0.264433],
              [0.275495, 0.187941, 0.251305, 0.347122],
              [0.382803, 0.390766, 0.363815, 0.289582]],
      
             [[0.248055, 0.482641, 0.316236, 0.123329],
              [0.237221, 0.493156, 0.360972, 0.172186],
              [0.249204, 0.202971, 0.194018, 0.607862],
              ...,
              [0.267913, 0.355198, 0.340389, 0.262352],
              [0.251883, 0.225029, 0.252392, 0.230649],
              [0.212175, 0.390106, 0.296799, 0.237791]],
      
             [[0.312122, 0.264439, 0.305117, 0.537302],
              [0.214007, 0.262056, 0.242397, 0.418294],
              [0.22629 , 0.212833, 0.331483, 0.218443],
              ...,
              [0.412739, 0.229345, 0.137388, 0.142437],
              [0.150303, 0.2786  , 0.399709, 0.514343],
              [0.29136 , 0.330588, 0.246156, 0.258651]],
      
             [[0.373154, 0.458734, 0.549744, 0.262994],
              [0.305288, 0.317411, 0.182924, 0.274919],
              [0.257765, 0.175467, 0.253729, 0.112986],
              ...,
              [0.320187, 0.373244, 0.253835, 0.27442 ],
              [0.374871, 0.319184, 0.226431, 0.284043],
              [0.152189, 0.173036, 0.247032, 0.201709]]])

    • gamma
      (chain, draw, strain)
      float64
      0.6667 0.4932 ... 0.6464 0.5471

      array([[[0.666749, 0.493181, 0.688938, 0.584822],
              [0.624827, 0.507767, 0.685517, 0.558233],
              [0.676103, 0.49344 , 0.635581, 0.592427],
              ...,
              [0.785935, 0.552467, 0.622423, 0.55091 ],
              [0.632281, 0.50102 , 0.577259, 0.634714],
              [0.68576 , 0.535868, 0.71355 , 0.630226]],
      
             [[0.583801, 0.538425, 0.66964 , 0.540686],
              [0.560697, 0.524561, 0.66983 , 0.588649],
              [0.627847, 0.484065, 0.615472, 0.601768],
              ...,
              [0.626668, 0.590898, 0.629342, 0.581877],
              [0.5824  , 0.459722, 0.650795, 0.556524],
              [0.665122, 0.566713, 0.611796, 0.5699  ]],
      
             [[0.77494 , 0.475923, 0.68726 , 0.657595],
              [0.687042, 0.556764, 0.699011, 0.581779],
              [0.461218, 0.440789, 0.710006, 0.518664],
              ...,
              [0.623083, 0.467364, 0.647316, 0.536071],
              [0.636356, 0.508061, 0.707323, 0.627238],
              [0.653192, 0.485772, 0.642366, 0.591617]],
      
             [[0.496705, 0.434849, 0.689312, 0.53787 ],
              [0.737713, 0.474942, 0.689566, 0.558079],
              [0.639783, 0.413196, 0.653105, 0.574101],
              ...,
              [0.637417, 0.510566, 0.664675, 0.592931],
              [0.727229, 0.477182, 0.652764, 0.558776],
              [0.625947, 0.41544 , 0.646447, 0.54707 ]]])

    • abundance
      (chain, draw, tube, timepoint, species)
      float64
      0.1236 1.159 ... 1.925 0.06352

      array([[[[[0.123611 , 1.15852  ],
                [0.137854 , 1.14902  ],
                [0.153712 , 1.13845  ],
                ...,
                [0.749507 , 0.741202 ],
                [0.826917 , 0.689588 ],
                [0.910472 , 0.633878 ]],
      
               [[0.139905 , 1.32972  ],
                [0.161601 , 1.31902  ],
                [0.186621 , 1.30668  ],
                ...,
                [1.47913  , 0.669234 ],
                [1.66813  , 0.576023 ],
                [1.86903  , 0.476944 ]],
      
               [[0.141784 , 1.2286   ],
                [0.165531 , 1.21224  ],
                [0.193156 , 1.19321  ],
                ...,
      ...
                ...,
                [1.12276  , 0.73846  ],
                [1.27587  , 0.674851 ],
                [1.44611  , 0.604127 ]],
      
               [[0.121848 , 1.22738  ],
                [0.14337  , 1.21347  ],
                [0.168636 , 1.19714  ],
                ...,
                [1.58111  , 0.284048 ],
                [1.73765  , 0.182852 ],
                [1.86409  , 0.101114 ]],
      
               [[0.141904 , 1.03885  ],
                [0.166155 , 1.02558  ],
                [0.19448  , 1.01008  ],
                ...,
                [1.67017  , 0.202776 ],
                [1.81562  , 0.123207 ],
                [1.92472  , 0.0635183]]]]])

  • Indexes: (10)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • ln_mu_max_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_mu_max_z_dim_0'))

    • ln_ks_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_ks_z_dim_0'))

    • ln_gamma_z_dim_0
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='ln_gamma_z_dim_0'))

    • tube
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16], dtype='int64', name='tube'))

    • species
      PandasIndex

      PandasIndex(Index([1, 2], dtype='int64', name='species'))

    • sigma_y_dim_0
      PandasIndex

      PandasIndex(Index([0, 1], dtype='int64', name='sigma_y_dim_0'))

    • strain
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4], dtype='int64', name='strain'))

    • timepoint
      PandasIndex

      PandasIndex(Index([1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20], dtype='int64', name='timepoint'))

  • Attributes: (4)

    created_at :

    2024-04-29T15:10:39.435568

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• posterior_predictive

  <xarray.Dataset> Size: 2MB
  Dimensions:      (chain: 4, draw: 300, measurement: 160)
  Coordinates:
    * chain        (chain) int64 32B 0 1 2 3
    * draw         (draw) int64 2kB 0 1 2 3 4 5 6 ... 293 294 295 296 297 298 299
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      yrep         (chain, draw, measurement) float64 2MB 0.165 1.264 ... 0.2912
  Attributes:
      created_at:                 2024-04-29T15:10:39.442620
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • yrep
      (chain, draw, measurement)
      float64
      0.165 1.264 0.2592 ... 1.474 0.2912

      array([[[0.165016, 1.26384 , 0.259152, ..., 0.540311, 1.50057 ,
               0.349396],
              [0.165817, 1.38195 , 0.269176, ..., 0.479713, 1.50554 ,
               0.282329],
              [0.176522, 1.22402 , 0.255909, ..., 0.544127, 1.40027 ,
               0.356926],
              ...,
              [0.158403, 1.35487 , 0.275877, ..., 0.593262, 1.40963 ,
               0.328979],
              [0.185599, 1.26854 , 0.258592, ..., 0.4823  , 1.53165 ,
               0.337027],
              [0.180958, 0.948152, 0.265112, ..., 0.494379, 1.55993 ,
               0.358883]],
      
             [[0.211212, 1.40198 , 0.248002, ..., 0.649566, 1.6177  ,
               0.40851 ],
              [0.204468, 1.23807 , 0.26564 , ..., 0.536478, 1.52677 ,
               0.301465],
              [0.161963, 1.21814 , 0.218207, ..., 0.613939, 1.61377 ,
               0.35846 ],
      ...
              [0.170971, 1.06513 , 0.244566, ..., 0.447323, 1.66761 ,
               0.325317],
              [0.186518, 1.56446 , 0.257964, ..., 0.575865, 1.62696 ,
               0.281266],
              [0.176738, 1.28539 , 0.245257, ..., 0.427258, 1.58304 ,
               0.319493]],
      
             [[0.172128, 1.12761 , 0.293089, ..., 0.575361, 1.61881 ,
               0.331317],
              [0.145485, 1.26374 , 0.218726, ..., 0.368051, 1.4473  ,
               0.293857],
              [0.170784, 1.19626 , 0.250898, ..., 0.535402, 1.70888 ,
               0.292073],
              ...,
              [0.161756, 1.34807 , 0.218135, ..., 0.456208, 1.79452 ,
               0.31489 ],
              [0.177366, 1.06993 , 0.234113, ..., 0.479718, 1.76014 ,
               0.260256],
              [0.195078, 1.0456  , 0.271717, ..., 0.50243 , 1.47396 ,
               0.291242]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-29T15:10:39.442620

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• log_likelihood

  <xarray.Dataset> Size: 2MB
  Dimensions:      (chain: 4, draw: 300, measurement: 160)
  Coordinates:
    * chain        (chain) int64 32B 0 1 2 3
    * draw         (draw) int64 2kB 0 1 2 3 4 5 6 ... 293 294 295 296 297 298 299
    * measurement  (measurement) int64 1kB 1 2 3 4 5 6 ... 155 156 157 158 159 160
  Data variables:
      llik         (chain, draw, measurement) float64 2MB 3.021 -0.1452 ... 0.8522
  Attributes:
      created_at:                 2024-04-29T15:10:39.443956
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
    • measurement: 160
  • Coordinates: (3)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

    • measurement
      (measurement)
      int64
      1 2 3 4 5 6 ... 156 157 158 159 160

      array([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,  11,  12,  13,  14,
              15,  16,  17,  18,  19,  20,  21,  22,  23,  24,  25,  26,  27,  28,
              29,  30,  31,  32,  33,  34,  35,  36,  37,  38,  39,  40,  41,  42,
              43,  44,  45,  46,  47,  48,  49,  50,  51,  52,  53,  54,  55,  56,
              57,  58,  59,  60,  61,  62,  63,  64,  65,  66,  67,  68,  69,  70,
              71,  72,  73,  74,  75,  76,  77,  78,  79,  80,  81,  82,  83,  84,
              85,  86,  87,  88,  89,  90,  91,  92,  93,  94,  95,  96,  97,  98,
              99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112,
             113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126,
             127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140,
             141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154,
             155, 156, 157, 158, 159, 160])

  • Data variables: (1)
    • llik
      (chain, draw, measurement)
      float64
      3.021 -0.1452 ... 0.7567 0.8522

      array([[[ 3.02052   , -0.145202  ,  2.98193   , ...,  0.142552  ,
                0.962963  ,  1.77857   ],
              [ 3.21125   ,  1.13259   ,  2.67426   , ...,  0.417259  ,
                0.748929  ,  1.29388   ],
              [ 3.09634   , -0.494584  ,  2.7348    , ..., -0.525809  ,
                0.474149  ,  2.08624   ],
              ...,
              [ 3.28388   ,  0.0766979 ,  2.87102   , ..., -0.273911  ,
                0.957292  ,  2.2776    ],
              [ 3.38554   ,  0.208544  ,  2.97293   , ...,  1.75768   ,
                0.940506  , -0.466631  ],
              [ 3.13574   ,  0.883512  ,  2.56507   , ..., -0.0327459 ,
                0.0904921 ,  2.19253   ]],
      
             [[ 3.31755   ,  0.331693  ,  2.85386   , ..., -1.73744   ,
                1.14065   ,  2.34866   ],
              [ 2.67885   ,  0.220435  ,  1.78304   , ...,  1.2344    ,
                0.716091  ,  0.152024  ],
              [ 3.15764   ,  0.619111  ,  2.98977   , ..., -0.463357  ,
                0.675184  ,  2.33388   ],
      ...
              [ 3.28578   , -0.460131  ,  2.99531   , ...,  0.463836  ,
                1.12539   ,  0.819732  ],
              [ 3.33478   ,  0.905078  ,  2.876     , ..., -0.105507  ,
                0.314407  ,  2.25966   ],
              [ 3.05777   , -0.201469  ,  3.03457   , ...,  0.832492  ,
                1.05823   ,  0.512679  ]],
      
             [[ 3.2593    , -0.905111  ,  2.64716   , ..., -0.595808  ,
                1.12041   ,  2.30362   ],
              [ 2.59617   ,  0.986292  ,  2.8081    , ...,  1.25099   ,
                0.820455  ,  1.10249   ],
              [ 2.92369   ,  0.684769  ,  2.81121   , ...,  0.925476  ,
                0.902426  ,  0.495239  ],
              ...,
              [ 3.27852   ,  0.880963  ,  3.09798   , ...,  1.75891   ,
                1.1329    , -0.825555  ],
              [ 3.10908   ,  0.486035  ,  3.12885   , ...,  1.73172   ,
                1.22233   , -0.520372  ],
              [ 2.91384   , -0.873578  ,  2.82839   , ...,  1.42908   ,
                0.756659  ,  0.852235  ]]])

  • Indexes: (3)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

    • measurement
      PandasIndex

      PandasIndex(Index([  1,   2,   3,   4,   5,   6,   7,   8,   9,  10,
             ...
             151, 152, 153, 154, 155, 156, 157, 158, 159, 160],
            dtype='int64', name='measurement', length=160))

  • Attributes: (4)

    created_at :

    2024-04-29T15:10:39.443956

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  
• sample_stats

  <xarray.Dataset> Size: 61kB
  Dimensions:          (chain: 4, draw: 300)
  Coordinates:
    * chain            (chain) int64 32B 0 1 2 3
    * draw             (draw) int64 2kB 0 1 2 3 4 5 6 ... 294 295 296 297 298 299
  Data variables:
      lp               (chain, draw) float64 10kB 99.26 93.29 ... 97.05 96.51
      acceptance_rate  (chain, draw) float64 10kB 0.9906 0.9414 ... 0.9219 0.9104
      step_size        (chain, draw) float64 10kB 0.02773 0.02773 ... 0.03921
      tree_depth       (chain, draw) int64 10kB 7 7 7 7 7 7 7 7 ... 7 7 7 6 6 7 7
      n_steps          (chain, draw) int64 10kB 127 127 255 127 ... 63 63 127 127
      diverging        (chain, draw) bool 1kB False False False ... False False
      energy           (chain, draw) float64 10kB -72.0 -68.86 ... -72.84 -68.01
  Attributes:
      created_at:                 2024-04-29T15:10:39.440206
      arviz_version:              0.17.0
      inference_library:          cmdstanpy
      inference_library_version:  1.2.1

  xarray.Dataset

  • Dimensions:
    • chain: 4
    • draw: 300
  • Coordinates: (2)
    • chain
      (chain)
      int64
      0 1 2 3

      array([0, 1, 2, 3])

    • draw
      (draw)
      int64
      0 1 2 3 4 5 ... 295 296 297 298 299

      array([  0,   1,   2, ..., 297, 298, 299])

  • Data variables: (7)
    • lp
      (chain, draw)
      float64
      99.26 93.29 102.4 ... 97.05 96.51

      array([[ 99.2575,  93.2898, 102.384 , ..., 102.501 ,  95.5903, 101.189 ],
             [ 98.3238, 104.506 , 104.146 , ..., 101.723 , 106.189 , 104.725 ],
             [107.548 , 105.395 ,  98.2402, ..., 102.119 ,  98.8123, 103.261 ],
             [ 85.507 ,  90.2359,  91.5547, ..., 100.316 ,  97.0465,  96.5065]])

    • acceptance_rate
      (chain, draw)
      float64
      0.9906 0.9414 ... 0.9219 0.9104

      array([[0.990616, 0.941424, 0.996313, ..., 0.997699, 0.94197 , 0.999361],
             [0.98955 , 0.880898, 0.914559, ..., 0.997897, 0.776587, 0.999823],
             [0.957117, 0.929493, 0.887482, ..., 0.979757, 0.973815, 0.940598],
             [0.747767, 0.986848, 0.827413, ..., 0.974095, 0.92186 , 0.910375]])

    • step_size
      (chain, draw)
      float64
      0.02773 0.02773 ... 0.03921 0.03921

      array([[0.0277288, 0.0277288, 0.0277288, ..., 0.0277288, 0.0277288,
              0.0277288],
             [0.0320218, 0.0320218, 0.0320218, ..., 0.0320218, 0.0320218,
              0.0320218],
             [0.0357054, 0.0357054, 0.0357054, ..., 0.0357054, 0.0357054,
              0.0357054],
             [0.0392062, 0.0392062, 0.0392062, ..., 0.0392062, 0.0392062,
              0.0392062]])

    • tree_depth
      (chain, draw)
      int64
      7 7 7 7 7 7 7 7 ... 7 7 7 7 6 6 7 7

      array([[7, 7, 7, ..., 7, 7, 7],
             [7, 7, 7, ..., 7, 7, 7],
             [7, 7, 7, ..., 7, 7, 7],
             [7, 7, 7, ..., 6, 7, 7]])

    • n_steps
      (chain, draw)
      int64
      127 127 255 127 ... 63 63 127 127

      array([[127, 127, 255, ..., 127, 127, 127],
             [127, 127, 127, ..., 127, 127, 127],
             [127, 127, 127, ..., 127, 127, 127],
             [127, 127, 127, ...,  63, 127, 127]])

    • diverging
      (chain, draw)
      bool
      False False False ... False False

      array([[False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False],
             [False, False, False, ..., False, False, False]])

    • energy
      (chain, draw)
      float64
      -72.0 -68.86 ... -72.84 -68.01

      array([[-71.9981, -68.86  , -73.0806, ..., -81.0142, -59.7193, -72.5441],
             [-81.4027, -69.601 , -83.8125, ..., -68.3654, -80.0367, -86.1678],
             [-86.8943, -81.5087, -73.2493, ..., -78.6303, -73.7353, -75.1187],
             [-48.115 , -52.8319, -62.716 , ..., -82.5446, -72.8391, -68.0126]])

  • Indexes: (2)
    • chain
      PandasIndex

      PandasIndex(Index([0, 1, 2, 3], dtype='int64', name='chain'))

    • draw
      PandasIndex

      PandasIndex(Index([  0,   1,   2,   3,   4,   5,   6,   7,   8,   9,
             ...
             290, 291, 292, 293, 294, 295, 296, 297, 298, 299],
            dtype='int64', name='draw', length=300))

  • Attributes: (4)

    created_at :

    2024-04-29T15:10:39.440206

    arviz_version :

    0.17.0

    inference_library :

    cmdstanpy

    inference_library_version :

    1.2.1

  
  

Diagnostics: is the posterior ok?

First check the sample_stats group to see if there were any divergent transitions and if the lp parameter converged.

az.summary(idata_posterior.sample_stats)

/Users/tedgro/repos/biosustain/bayesian_statistics_for_computational_biology/.venv/lib/python3.12/site-packages/arviz/stats/diagnostics.py:592: RuntimeWarning: invalid value encountered in scalar divide
  (between_chain_variance / within_chain_variance + num_samples - 1) / (num_samples)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
lp	99.182	5.730	88.560	109.161	0.288	0.204	413.0	634.0	1.010000e+00
acceptance_rate	0.936	0.079	0.783	1.000	0.002	0.002	1308.0	1111.0	1.010000e+00
step_size	0.034	0.004	0.028	0.039	0.002	0.002	4.0	4.0	5.859337e+15
tree_depth	6.966	0.211	7.000	7.000	0.022	0.016	91.0	947.0	1.050000e+00
n_steps	131.213	27.896	63.000	127.000	4.238	3.017	49.0	53.0	1.070000e+00
diverging	0.000	0.000	0.000	0.000	0.000	0.000	1200.0	1200.0	NaN
energy	-72.876	7.834	-87.030	-58.601	0.386	0.274	413.0	730.0	1.010000e+00

Next check the parameter-by-parameter summary

az.summary(idata_posterior)

	mean	sd	hdi_3%	hdi_97%	mcse_mean	mcse_sd	ess_bulk	ess_tail	r_hat
ln_mu_max_z[0]	-0.787	0.516	-1.744	0.127	0.020	0.014	661.0	798.0	1.01
ln_mu_max_z[1]	0.402	0.509	-0.533	1.364	0.020	0.014	624.0	773.0	1.00
ln_mu_max_z[2]	0.777	0.513	-0.230	1.665	0.021	0.015	586.0	730.0	1.01
ln_mu_max_z[3]	0.886	0.518	-0.050	1.884	0.021	0.015	607.0	719.0	1.00
ln_ks_z[0]	-0.009	1.006	-1.807	1.813	0.024	0.029	1733.0	951.0	1.00
...	...	...	...	...	...	...	...	...	...
abundance[16, 18, 2]	0.229	0.027	0.183	0.283	0.001	0.001	1476.0	1055.0	1.00
abundance[16, 19, 1]	1.799	0.087	1.648	1.981	0.002	0.001	1999.0	1080.0	1.00
abundance[16, 19, 2]	0.148	0.026	0.097	0.197	0.001	0.001	1276.0	1061.0	1.00
abundance[16, 20, 1]	1.909	0.096	1.733	2.100	0.002	0.002	1864.0	1087.0	1.00
abundance[16, 20, 2]	0.085	0.024	0.038	0.125	0.001	0.000	1188.0	758.0	1.00

704 rows × 9 columns

Show posterior intervals

prior_abundances = idata_posterior.posterior["abundance"]

n_sample = 20
chains = rng.choice(prior_abundances.coords["chain"].values, n_sample)
draws = rng.choice(prior_abundances.coords["draw"].values, n_sample)
f, axes = plot_sim(true_abundance, fake_measurements, species_to_ax)

for ax, species_i in zip(axes, species):
    for tube_j in tubes:
        for chain, draw in zip(chains, draws):
            timeseries = prior_abundances.sel(chain=chain, draw=draw, tube=tube_j, species=species_i)
            ax.plot(
                timepoints.values, 
                timeseries.values,
                alpha=0.5, color="skyblue", zorder=-1
            )
f.savefig("img/monod_posteriors.png")

look at the posterior

The next few cells use arviz’s plot_posterior function to plot the marginal posterior distributions for some of the model’s parameters:

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["gamma"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["gamma"]):
    ax.axvline(true_value, color="red")

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["mu_max"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["mu_max"]):
    ax.axvline(true_value, color="red")

f, axes = plt.subplots(1, 4, figsize=[10, 4])
axes = az.plot_posterior(
    idata_posterior,
    kind="hist",
    bins=20,
    var_names=["ks"],
    ax=axes,
    point_estimate=None,
    hdi_prob="hide"
)
for ax, true_value in zip(axes, true_param_values["ks"]):
    ax.axvline(true_value, color="red")

References

Allen, Rosalind J, and Bartłomiej Waclaw. 2019. “Bacterial Growth: A Statistical Physicist’s Guide.” Reports on Progress in Physics. Physical Society (Great Britain) 82 (1): 016601. https://doi.org/10.1088/1361-6633/aae546.

Timonen, Juho, Nikolas Siccha, Ben Bales, Harri Lähdesmäki, and Aki Vehtari. 2022. “An Importance Sampling Approach for Reliable and Efficient Inference in Bayesian Ordinary Differential Equation Models.” arXiv. https://doi.org/10.48550/arXiv.2205.09059.